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Generators of the pro-p Iwahori and Galois representations

2017-06-23
Christophe Cornut , Jishnu Ray
For an odd prime [Formula: see text], we determine a minimal set of topological generators of the pro-[Formula: see text] Iwahori subgroup of a split reductive group [Formula: see text] … For an odd prime [Formula: see text], we determine a minimal set of topological generators of the pro-[Formula: see text] Iwahori subgroup of a split reductive group [Formula: see text] over [Formula: see text]. In the simple adjoint case and for any sufficiently large regular prime [Formula: see text], we also construct Galois extensions of [Formula: see text] with Galois group between the pro-[Formula: see text] and the standard Iwahori subgroups of [Formula: see text].
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Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>

2018-06-30
Jishnu Ray

Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's p-rationality conjecture

2019-01-01
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational … In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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SOME REMARKS AND EXPERIMENTS ON GREENBERG'S p-RATIONALITY CONJECTURE

2017-06-07
Razvan Barbulescu , Jishnu Ray
The object of this article is to discuss a conjecture of Greenberg and its links to the Galois inverse problem. We show that it is related to well established conjectures … The object of this article is to discuss a conjecture of Greenberg and its links to the Galois inverse problem. We show that it is related to well established conjectures in algebraic number theory and that some particular cases are corollaries of known results. Finally, we do numerical experiments which allow to formulate new conjectures which imply Greenberg's conjecture.

Explicit ring-theoretic presentation of Iwasawa algebras

2018-10-30
Jishnu Ray
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Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture

2020-08-21
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg’s p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational … In this paper we make a series of numerical experiments to support Greenberg’s p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen–Lenstra–Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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Explicit presentation of an Iwasawa algebra: The case of pro-<i>p</i> Iwahori subgroup of SL<sub> <i>n</i> </sub>(ℤ<sub> <i>p</i> </sub>)

2019-11-08
Jishnu Ray
Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory … Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p -adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} which were uniform pro- p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {p&gt;n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro- p Iwahori subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro- p group.

Globally analytic principal series representation and Langlands base change

2020-09-04
Jishnu Ray
S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition … S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. In this article, we take the case of $GL(n)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL(n,\mathbb{Z}_p)$, following the notion of globally analytic representations introduced by M. Emerton. Furthermore, we relate the condition of irreducibility of our globally analytic principal series to that of a Verma module. Finally, using Steinberg tensor product theorem, we construct Langlands base change of our globally analytic principal series to a finite unramified extension of $\mathbb{Q}_p$, generalizing an earlier work of Clozel for $GL(2)$.
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Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-09-11
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.

Presentation of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$

2017-07-21
Jishnu Ray
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic … Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.

Selmer groups of elliptic curves over the $PGL(2)$ extension

2020-01-01
Jishnu Ray , R. Sujatha
Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension … Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension to study the dual Selmer group over the $PGL(2)$ extension.

Presentation of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$

2017-01-01
Jishnu Ray
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic … Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.
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Rigid analytic vectors of crystalline representations arising in p-adic Langlands

2021-09-21
Jishnu Ray
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Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

2022-04-08
Jeffrey Hatley , Debanjana Kundu , Antonio Lei +1 more
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Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's $p$-rationality conjecture

2017-01-01
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational … In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-01-01
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.
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A formal model of Coleman families and applications to Iwasawa invariants

2023-01-01
Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio , Tadashi Ochiai , Jishnu Ray
For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we … For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we study the variations of Iwasawa $\lambda$- and $\mu$-invariants of dualfine (strict) Selmer groups over the cyclotomic Zp-extension of Q in Coleman families ofmodular forms. This generalizes an earlier work of Jha and Sujatha for Hida families.
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Iwasawa algebras for p-adic Lie groups and Galois groups

2018-07-02
Jishnu Ray
A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in … A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in varied settings such as Lazard's work on p-adic Lie groups and Fontaine's work on local Galois representations. For a prime p, the Iwasawa algebra of a p-adic Lie group G, is a non-commutative completed group algebra of G which is also the algebra of p-adic measures on G. It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups as noticed by Clozel. In Part I, we lay the foundation by giving an explicit description of certain Iwasawa algebras. We first find an explicit presentation (by generators and relations) of the Iwasawa algebra for the principal congruence subgroup of any semi-simple, simply connected Chevalley group over Z_p. Furthermore, we extend the method to give a set of generators and relations for the Iwasawa algebra of the pro-p Iwahori subgroup of GL(n,Z_p). The base change map between the Iwasawa algebras over an extension of Q_p motivates us to study the globally analytic p-adic representations following Emerton's work. We also provide results concerning the globally analytic induced principal series representation under the action of the pro-p Iwahori subgroup of GL(n,Z_p) and determine its condition of irreducibility. In Part II, we do numerical experiments using a computer algebra system SAGE which give heuristic support to Greenberg's p-rationality conjecture affirming the existence of number fields with Galois groups (Z/2Z)^t. The p-rational fields are algebraic number fields whose Galois cohomology is particularly simple and they offer ways of constructing Galois representations with big open images. We go beyond Greenberg's work and construct new Galois representations of the absolute Galois group of Q with big open images in reductive groups over Z_p (ex. GL(n, Z_p), SL(n, Z_p), SO(n, Z_p), Sp(2n, Z_p)). We are proving results which show the existence of p-adic Lie extensions of Q where the Galois group corresponds to a certain specific p-adic Lie algebra (ex. sl(n), so(n), sp(2n)). This relates our work with a more general and classical inverse Galois problem for p-adic Lie extensions.
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Generators of the pro-p Iwahori and Galois representations

2016-11-18
Christophe Cornut , Jishnu Ray
For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint … For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p and the standard Iwahori subgroups of G.

Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

2019-01-01
Dong Han , Jishnu Ray , Feng Wei
For a prime $p&gt;2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $Ω_{G(1)}$ be the mod-$p$ Iwasawa … For a prime $p&gt;2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $Ω_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field $\mathbb{F}_p$. Ardakov, Wei, Zhang have shown that if $p$ is a "nice prime " ($p \geq 5$ and $p \nmid n+1$ if the Lie algebra of $G(1)$ is of type $A_n$), then every non-zero normal element in $Ω_{G(1)}$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture.

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

2021-12-10
Jishnu Ray , Feng Wei , Gergely Zábrádi
Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α .Let ∆ be a finite set.We construct an equivalence of categories … Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α .Let ∆ be a finite set.We construct an equivalence of categories between finite dimensional F p -representations of the product of ∆ copies of the absolute Galois group of K and multivariable étale (ϕ, Γ)-modules over a multivariable Laurent series ring over k α .Résumé ((ϕ, Γ)-modules multivariables et représentations du produit du groupe de Galois: le cas des corps résiduels imparfaits) Soit K un corps discrètement valué à charactéristique mixte (0, p) et un corps résiduel imparfait k α .Soit ∆ un ensemble fini.Nous établissons une équivalence de catégories entre des représentations de dimensions finies sur F p du produit de ∆ copies du groupe absolu de Galois de K et des (ϕ, Γ)-modules étales multivariables sur un anneau multivariable des séries Laurent sur k α .
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On the growth of $\mu $-invariant in Iwasawa theory of supersingular elliptic curves

2022-01-01
Jishnu Ray
In this article, we provide a relation between the µ-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic Z p … In this article, we provide a relation between the µ-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic Z p -extension to a Z 2 p -extension over an imaginary quadratic field.Furthermore we show that the (supersingular) M H (G)-conjecture is equivalent to the fact that the µ-invariant doesn't change as we go up the tower.
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Normal elements in the Iwasawa algebras of Chevalley groups

2020-06-16
Dong Han , Jishnu Ray , Feng Wei

Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

2020-06-24
Jishnu Ray
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the … Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\mathbf{B}(V)_{\mathrm{la}}$. In particular, we show the existence of rigid analytic vectors inside $\mathbf{B}(V)_{\mathrm{la}}$ and prove that its non-null. This gives us a rigid analytic representation (in the sense of Emerton) lying inside the locally analytic representation $\mathbf{B}(V)_{\mathrm{la}}$.

Conjecture A and $\mu$-invariant for Selmer groups of supersingular elliptic curves

2020-06-25
Parham Hamidi , Jishnu Ray
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, … Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

2020-01-01
Antonio Lei , Jishnu Ray
Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed … Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $\Pi$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $\Pi$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\mathbb{Q}$ attached to the $p$-adic Galois representation of $\Pi$ and formulate Iwasawa main conjectures in the spirit of Kobayashi's plus and minus main conjectures for $p$-supersingular elliptic curves.
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On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-12-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.

Conjecture A and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math>-invariant for Selmer groups of supersingular elliptic curves

2022-01-26
Parham Hamidi , Jishnu Ray
Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey … Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ℤ p 2 -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the μ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when E has good supersingular reduction, which are completely analogous to the classical Selmer group, when E has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.
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Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

2022-01-01
Jeffrey Hatley , Debanjana Kundu , Antonio Lei +1 more
Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We … Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We apply these control theorems to compare the fine Selmer group attached to a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ to its counterpart attached to the conjugate modular form $\overline{f}$.

On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-12-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.
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On characteristic power series of dual signed Selmer groups

2022-01-01
Jishnu Ray , Florian Sprung
We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of … We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of the characteristic power series of the signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. This generalizes a result of Vigni and Longo in the ordinary case. In the case of elliptic curves, such results follow from earlier works by Greenberg, Kim, the second author, and Ahmed-Lim, covering both the ordinary and most of the supersingular case.
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SELMER GROUPS OF ELLIPTIC CURVES OVER THE EXTENSION

2022-05-30
Jishnu Ray , R. Sujatha
Abstract Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p -adic Lie extension, there exists a structure theorem … Abstract Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p -adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

2020-01-01
Parham Hamidi , Jishnu Ray
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, … Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.
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Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

2020-01-01
Jishnu Ray
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the … Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\mathbf{B}(V)_{\mathrm{la}}$. In particular, we show the existence of rigid analytic vectors inside $\mathbf{B}(V)_{\mathrm{la}}$ and prove that its non-null. This gives us a rigid analytic representation (in the sense of Emerton) lying inside the locally analytic representation $\mathbf{B}(V)_{\mathrm{la}}$.
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On the growth of $μ$-invariant in Iwasawa theory of supersingular Elliptic curves

2020-01-01
Jishnu Ray
In this article, we provide a relation between the $\mu$-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic $\mathbb{Z}_p$-extension to … In this article, we provide a relation between the $\mu$-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic $\mathbb{Z}_p$-extension to a $\mathbb{Z}_p^2$-extension over an imaginary quadratic field. Furthermore we show that the supersingular $\mathfrak{M}_H(G)$-conjecture is equivalent to the fact that the $\mu$-invariants doesn't change as we go up the tower.

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-09-11
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.
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Generators of the pro-p Iwahori and Galois representations

2016-01-01
Christophe Cornut , Jishnu Ray
For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint … For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p and the standard Iwahori subgroups of G.
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On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-01-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.

Iwasawa theory of automorphic representations of $$\textrm{GL}_{2n}$$ at non-ordinary primes

2022-11-22
Antonio Lei , Jishnu Ray

Presentation of an Iwasawa algebra:The pro-$p$-Iwahori of reductive groups

2023-01-01
Aranya Lahiri , Jishnu Ray
In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations … In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of a connected, split, reductive group $\mathbb{G}$ over $\mathbb{Q}_p$.

Asymptotic growth of the signed Tate-Shafarevich groups for supersingular abelian varieties

2023-01-01
Jishnu Ray
Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei … Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. In this paper, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Buyukboduk--Lei, we define the multi-signed Mordell-Weil groups for supersingular abelian varieties, provide an explicit structure of the dual of these groups as an Iwasawa module and prove a control theorem. Furthermore, we define the multi-signed Tate-Shafarevich groups and, by computing their Kobayashi rank, we provide an asymptotic growth formula along the cyclotomic tower of $\mathbb{Q}$.
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On the signed Selmer groups for motives at non-ordinary primes in $\mathbb{Z}_p^2$-extensions

2023-01-01
Jishnu Ray , Florian Sprung
Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension … Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary motives. In particular, their construction applies to abelian varieties over $F$ with good supersingular reduction at all the primes of $F$ above $p$. In this article, we scrutinize the case in which $F$ is imaginary quadratic, and prove a control theorem (that generalizes Kim's control theorem for elliptic curves) of multi-signed Selmer groups of non-ordinary motives over the maximal abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the $\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition when these multi-signed Selmer groups are cotorsion over the corresponding two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa $\mu$-invariants of multi-signed Selmer groups over the $\mathbb{Z}_p^2$-extension for two such representations which are congruent modulo $p$.
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Characteristic ideal of the fine Selmer group and results on $\mu$-invariance under isogeny in the function field case

2024-06-05
Sohan Ghosh , Jishnu Ray
Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. … Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we study the algebraic structure and prove a control theorem for the S-fine Mordell-Weil groups, the function field analogue for Wuthrich's fine Mordell-Weil groups, over a $\mathbb{Z}_p$-extension of $K$. In case of unramified $\mathbb{Z}_p$-extension, $K_\infty$, we compute the characteristic ideal of the Pontryagin dual of the S-fine Mordell group. This provides an answer to an analogue of Greenberg's question for the characteristic ideal of the dual fine Selmer group in the function field setup. In the $\ell\neq p$ case, we prove the triviality of the $\mu$-invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending Conjecture A. In the $\ell=p$ case, we compute the change of $\mu$-invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the $\mu$-invariant.
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On $\mu$-invariants and isogenies for abelian varieties over function fields

2024-07-31
Sohan Ghosh , Jishnu Ray , Takashi Suzuki
We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field … We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field setting. We also prove that the validity of the Birch--Swinnerton-Dyer conjecture (including the leading coefficient formula) over function fields is invariant under isogeny, without using the result of Kato--Trihan.
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Finiteness and cofiniteness of fine Selmer groups over function fields

2024-08-13
Sohan Ghosh , Jishnu Ray , Takashi Suzuki
We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field … We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable $p$-primary torsion of the abelian variety is finite (respectively zero).
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Presentation of an Iwasawa algebra: The pro-<i>p</i> Iwahori of simple, simply connected, split groups

2025-01-09
Aranya Lahiri , Jishnu Ray
Abstract In this article, we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ , respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators … Abstract In this article, we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ , respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro- p Iwahori subgroup of a simple, simply connected, split group $\mathbf {G}$ over ${{\mathbb Q}_p}$ .

On characteristic power series of dual signed Selmer groups

2025-02-10
Jishnu Ray , Florian Sprung

On the signed Selmer groups for motives at non-ordinary primes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>-extensions

2025-05-01
Jishnu Ray , Florian Sprung

On the signed Selmer groups for motives at non-ordinary primes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math>-extensions

2025-05-01
Jishnu Ray , Florian Sprung

On characteristic power series of dual signed Selmer groups

2025-02-10
Jishnu Ray , Florian Sprung

Presentation of an Iwasawa algebra: The pro-<i>p</i> Iwahori of simple, simply connected, split groups

2025-01-09
Aranya Lahiri , Jishnu Ray
Abstract In this article, we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ , respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators … Abstract In this article, we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ , respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro- p Iwahori subgroup of a simple, simply connected, split group $\mathbf {G}$ over ${{\mathbb Q}_p}$ .

Finiteness and cofiniteness of fine Selmer groups over function fields

2024-08-13
Sohan Ghosh , Jishnu Ray , Takashi Suzuki
We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field … We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable $p$-primary torsion of the abelian variety is finite (respectively zero).
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On $\mu$-invariants and isogenies for abelian varieties over function fields

2024-07-31
Sohan Ghosh , Jishnu Ray , Takashi Suzuki
We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field … We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field setting. We also prove that the validity of the Birch--Swinnerton-Dyer conjecture (including the leading coefficient formula) over function fields is invariant under isogeny, without using the result of Kato--Trihan.
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Characteristic ideal of the fine Selmer group and results on $\mu$-invariance under isogeny in the function field case

2024-06-05
Sohan Ghosh , Jishnu Ray
Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. … Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we study the algebraic structure and prove a control theorem for the S-fine Mordell-Weil groups, the function field analogue for Wuthrich's fine Mordell-Weil groups, over a $\mathbb{Z}_p$-extension of $K$. In case of unramified $\mathbb{Z}_p$-extension, $K_\infty$, we compute the characteristic ideal of the Pontryagin dual of the S-fine Mordell group. This provides an answer to an analogue of Greenberg's question for the characteristic ideal of the dual fine Selmer group in the function field setup. In the $\ell\neq p$ case, we prove the triviality of the $\mu$-invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending Conjecture A. In the $\ell=p$ case, we compute the change of $\mu$-invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the $\mu$-invariant.
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Presentation of an Iwasawa algebra:The pro-$p$-Iwahori of reductive groups

2023-01-01
Aranya Lahiri , Jishnu Ray
In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations … In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of a connected, split, reductive group $\mathbb{G}$ over $\mathbb{Q}_p$.

A formal model of Coleman families and applications to Iwasawa invariants

2023-01-01
Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio , Tadashi Ochiai , Jishnu Ray
For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we … For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we study the variations of Iwasawa $\lambda$- and $\mu$-invariants of dualfine (strict) Selmer groups over the cyclotomic Zp-extension of Q in Coleman families ofmodular forms. This generalizes an earlier work of Jha and Sujatha for Hida families.
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Asymptotic growth of the signed Tate-Shafarevich groups for supersingular abelian varieties

2023-01-01
Jishnu Ray
Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei … Let $E$ be an elliptic curve over $\mathbb{Q}$ with supersingular reduction at $p$ with $a_p=0$. We study the asymptotic growth of the plus and minus Tate-Shafarevich groups defined by Lei along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. In this paper, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Buyukboduk--Lei, we define the multi-signed Mordell-Weil groups for supersingular abelian varieties, provide an explicit structure of the dual of these groups as an Iwasawa module and prove a control theorem. Furthermore, we define the multi-signed Tate-Shafarevich groups and, by computing their Kobayashi rank, we provide an asymptotic growth formula along the cyclotomic tower of $\mathbb{Q}$.
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On the signed Selmer groups for motives at non-ordinary primes in $\mathbb{Z}_p^2$-extensions

2023-01-01
Jishnu Ray , Florian Sprung
Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension … Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary motives. In particular, their construction applies to abelian varieties over $F$ with good supersingular reduction at all the primes of $F$ above $p$. In this article, we scrutinize the case in which $F$ is imaginary quadratic, and prove a control theorem (that generalizes Kim's control theorem for elliptic curves) of multi-signed Selmer groups of non-ordinary motives over the maximal abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the $\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition when these multi-signed Selmer groups are cotorsion over the corresponding two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa $\mu$-invariants of multi-signed Selmer groups over the $\mathbb{Z}_p^2$-extension for two such representations which are congruent modulo $p$.
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Iwasawa theory of automorphic representations of $$\textrm{GL}_{2n}$$ at non-ordinary primes

2022-11-22
Antonio Lei , Jishnu Ray

SELMER GROUPS OF ELLIPTIC CURVES OVER THE EXTENSION

2022-05-30
Jishnu Ray , R. Sujatha
Abstract Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p -adic Lie extension, there exists a structure theorem … Abstract Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p -adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

2022-04-08
Jeffrey Hatley , Debanjana Kundu , Antonio Lei +1 more
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Conjecture A and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math>-invariant for Selmer groups of supersingular elliptic curves

2022-01-26
Parham Hamidi , Jishnu Ray
Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey … Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ℤ p 2 -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the μ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when E has good supersingular reduction, which are completely analogous to the classical Selmer group, when E has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.
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On the growth of $\mu $-invariant in Iwasawa theory of supersingular elliptic curves

2022-01-01
Jishnu Ray
In this article, we provide a relation between the µ-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic Z p … In this article, we provide a relation between the µ-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic Z p -extension to a Z 2 p -extension over an imaginary quadratic field.Furthermore we show that the (supersingular) M H (G)-conjecture is equivalent to the fact that the µ-invariant doesn't change as we go up the tower.
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Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms

2022-01-01
Jeffrey Hatley , Debanjana Kundu , Antonio Lei +1 more
Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We … Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. We prove two control theorems for fine Selmer groups of general cofinitely generated modules over $\mathcal{O}$. We apply these control theorems to compare the fine Selmer group attached to a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ to its counterpart attached to the conjugate modular form $\overline{f}$.

On characteristic power series of dual signed Selmer groups

2022-01-01
Jishnu Ray , Florian Sprung
We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of … We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of the characteristic power series of the signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. This generalizes a result of Vigni and Longo in the ordinary case. In the case of elliptic curves, such results follow from earlier works by Greenberg, Kim, the second author, and Ahmed-Lim, covering both the ordinary and most of the supersingular case.
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Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

2021-12-10
Jishnu Ray , Feng Wei , Gergely Zábrádi
Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α .Let ∆ be a finite set.We construct an equivalence of categories … Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α .Let ∆ be a finite set.We construct an equivalence of categories between finite dimensional F p -representations of the product of ∆ copies of the absolute Galois group of K and multivariable étale (ϕ, Γ)-modules over a multivariable Laurent series ring over k α .Résumé ((ϕ, Γ)-modules multivariables et représentations du produit du groupe de Galois: le cas des corps résiduels imparfaits) Soit K un corps discrètement valué à charactéristique mixte (0, p) et un corps résiduel imparfait k α .Soit ∆ un ensemble fini.Nous établissons une équivalence de catégories entre des représentations de dimensions finies sur F p du produit de ∆ copies du groupe absolu de Galois de K et des (ϕ, Γ)-modules étales multivariables sur un anneau multivariable des séries Laurent sur k α .
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On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-12-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.

On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-12-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.
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Rigid analytic vectors of crystalline representations arising in p-adic Langlands

2021-09-21
Jishnu Ray
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On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

2021-01-01
Cédric Dion , Jishnu Ray
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular … Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.

Globally analytic principal series representation and Langlands base change

2020-09-04
Jishnu Ray
S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition … S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. In this article, we take the case of $GL(n)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL(n,\mathbb{Z}_p)$, following the notion of globally analytic representations introduced by M. Emerton. Furthermore, we relate the condition of irreducibility of our globally analytic principal series to that of a Verma module. Finally, using Steinberg tensor product theorem, we construct Langlands base change of our globally analytic principal series to a finite unramified extension of $\mathbb{Q}_p$, generalizing an earlier work of Clozel for $GL(2)$.
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Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture

2020-08-21
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg’s p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational … In this paper we make a series of numerical experiments to support Greenberg’s p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen–Lenstra–Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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Conjecture A and $\mu$-invariant for Selmer groups of supersingular elliptic curves

2020-06-25
Parham Hamidi , Jishnu Ray
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, … Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.

Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

2020-06-24
Jishnu Ray
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the … Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\mathbf{B}(V)_{\mathrm{la}}$. In particular, we show the existence of rigid analytic vectors inside $\mathbf{B}(V)_{\mathrm{la}}$ and prove that its non-null. This gives us a rigid analytic representation (in the sense of Emerton) lying inside the locally analytic representation $\mathbf{B}(V)_{\mathrm{la}}$.

Normal elements in the Iwasawa algebras of Chevalley groups

2020-06-16
Dong Han , Jishnu Ray , Feng Wei

Selmer groups of elliptic curves over the $PGL(2)$ extension

2020-01-01
Jishnu Ray , R. Sujatha
Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension … Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension to study the dual Selmer group over the $PGL(2)$ extension.

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

2020-01-01
Antonio Lei , Jishnu Ray
Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed … Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $\Pi$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $\Pi$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\mathbb{Q}$ attached to the $p$-adic Galois representation of $\Pi$ and formulate Iwasawa main conjectures in the spirit of Kobayashi's plus and minus main conjectures for $p$-supersingular elliptic curves.
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Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

2020-01-01
Parham Hamidi , Jishnu Ray
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, … Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.
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Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

2020-01-01
Jishnu Ray
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the … Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\mathbf{B}(V)_{\mathrm{la}}$. In particular, we show the existence of rigid analytic vectors inside $\mathbf{B}(V)_{\mathrm{la}}$ and prove that its non-null. This gives us a rigid analytic representation (in the sense of Emerton) lying inside the locally analytic representation $\mathbf{B}(V)_{\mathrm{la}}$.
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On the growth of $μ$-invariant in Iwasawa theory of supersingular Elliptic curves

2020-01-01
Jishnu Ray
In this article, we provide a relation between the $\mu$-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic $\mathbb{Z}_p$-extension to … In this article, we provide a relation between the $\mu$-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic $\mathbb{Z}_p$-extension to a $\mathbb{Z}_p^2$-extension over an imaginary quadratic field. Furthermore we show that the supersingular $\mathfrak{M}_H(G)$-conjecture is equivalent to the fact that the $\mu$-invariants doesn't change as we go up the tower.

Explicit presentation of an Iwasawa algebra: The case of pro-<i>p</i> Iwahori subgroup of SL<sub> <i>n</i> </sub>(ℤ<sub> <i>p</i> </sub>)

2019-11-08
Jishnu Ray
Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory … Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p -adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} which were uniform pro- p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {p&gt;n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro- p Iwahori subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro- p group.

Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

2019-01-01
Dong Han , Jishnu Ray , Feng Wei
For a prime $p&gt;2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $Ω_{G(1)}$ be the mod-$p$ Iwasawa … For a prime $p&gt;2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $Ω_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field $\mathbb{F}_p$. Ardakov, Wei, Zhang have shown that if $p$ is a "nice prime " ($p \geq 5$ and $p \nmid n+1$ if the Lie algebra of $G(1)$ is of type $A_n$), then every non-zero normal element in $Ω_{G(1)}$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture.

Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's p-rationality conjecture

2019-01-01
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational … In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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Explicit ring-theoretic presentation of Iwasawa algebras

2018-10-30
Jishnu Ray
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Iwasawa algebras for p-adic Lie groups and Galois groups

2018-07-02
Jishnu Ray
A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in … A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in varied settings such as Lazard's work on p-adic Lie groups and Fontaine's work on local Galois representations. For a prime p, the Iwasawa algebra of a p-adic Lie group G, is a non-commutative completed group algebra of G which is also the algebra of p-adic measures on G. It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups as noticed by Clozel. In Part I, we lay the foundation by giving an explicit description of certain Iwasawa algebras. We first find an explicit presentation (by generators and relations) of the Iwasawa algebra for the principal congruence subgroup of any semi-simple, simply connected Chevalley group over Z_p. Furthermore, we extend the method to give a set of generators and relations for the Iwasawa algebra of the pro-p Iwahori subgroup of GL(n,Z_p). The base change map between the Iwasawa algebras over an extension of Q_p motivates us to study the globally analytic p-adic representations following Emerton's work. We also provide results concerning the globally analytic induced principal series representation under the action of the pro-p Iwahori subgroup of GL(n,Z_p) and determine its condition of irreducibility. In Part II, we do numerical experiments using a computer algebra system SAGE which give heuristic support to Greenberg's p-rationality conjecture affirming the existence of number fields with Galois groups (Z/2Z)^t. The p-rational fields are algebraic number fields whose Galois cohomology is particularly simple and they offer ways of constructing Galois representations with big open images. We go beyond Greenberg's work and construct new Galois representations of the absolute Galois group of Q with big open images in reductive groups over Z_p (ex. GL(n, Z_p), SL(n, Z_p), SO(n, Z_p), Sp(2n, Z_p)). We are proving results which show the existence of p-adic Lie extensions of Q where the Galois group corresponds to a certain specific p-adic Lie algebra (ex. sl(n), so(n), sp(2n)). This relates our work with a more general and classical inverse Galois problem for p-adic Lie extensions.
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Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>

2018-06-30
Jishnu Ray

Presentation of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$

2017-07-21
Jishnu Ray
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic … Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.

Generators of the pro-p Iwahori and Galois representations

2017-06-23
Christophe Cornut , Jishnu Ray
For an odd prime [Formula: see text], we determine a minimal set of topological generators of the pro-[Formula: see text] Iwahori subgroup of a split reductive group [Formula: see text] … For an odd prime [Formula: see text], we determine a minimal set of topological generators of the pro-[Formula: see text] Iwahori subgroup of a split reductive group [Formula: see text] over [Formula: see text]. In the simple adjoint case and for any sufficiently large regular prime [Formula: see text], we also construct Galois extensions of [Formula: see text] with Galois group between the pro-[Formula: see text] and the standard Iwahori subgroups of [Formula: see text].
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SOME REMARKS AND EXPERIMENTS ON GREENBERG'S p-RATIONALITY CONJECTURE

2017-06-07
Razvan Barbulescu , Jishnu Ray
The object of this article is to discuss a conjecture of Greenberg and its links to the Galois inverse problem. We show that it is related to well established conjectures … The object of this article is to discuss a conjecture of Greenberg and its links to the Galois inverse problem. We show that it is related to well established conjectures in algebraic number theory and that some particular cases are corollaries of known results. Finally, we do numerical experiments which allow to formulate new conjectures which imply Greenberg's conjecture.

Presentation of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$

2017-01-01
Jishnu Ray
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic … Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.
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Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's $p$-rationality conjecture

2017-01-01
Razvan Barbulescu , Jishnu Ray
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational … In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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Generators of the pro-p Iwahori and Galois representations

2016-11-18
Christophe Cornut , Jishnu Ray
For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint … For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p and the standard Iwahori subgroups of G.

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-09-11
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-09-11
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.
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Generators of the pro-p Iwahori and Galois representations

2016-01-01
Christophe Cornut , Jishnu Ray
For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint … For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p and the standard Iwahori subgroups of G.
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Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

2016-01-01
Jishnu Ray
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In … It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.
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Coauthor Papers Together
Razvan Barbulescu 4
Antonio Lei 4
Feng Wei 3
Parham Hamidi 3
Christophe Cornut 3
Cédric Dion 3
Jeffrey Hatley 2
R. Sujatha 2
Aranya Lahiri 2
Debanjana Kundu 2
Dong Han 2
Florian Sprung 2
Takashi Suzuki 2
Florian Sprung 2
Tadashi Ochiai 1
Sohan Ghosh 1
Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio 1
Sohan Ghosh 1
Gergely Zábrádi 1
Sohan Ghosh 1

Iwasawa theory for elliptic curves at supersingular primes

2003-04-01
Shin-ichi Kobayashi

On the structure theory of the Iwasawa algebra of a p-adic Lie group

2002-09-30
Otmar Venjakob
This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, 6 of a … This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, 6 of a p-adic analytic group G. For G without any p-torsion element we prove that 6 is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null 6-module. This is classical when G=Êkp for some integer kS1, but was previously unknown in the non-commutative case. Then the category of 6-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the Êp-torsion part of a finitely generated 6-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.
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Algebras of p-adic distributions and admissible representations

2003-07-01
Peter Schneider , Jeremy Teitelbaum
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Groupes analytiques $p$-adiques

1965-01-01
Michel Lazard

p-Adic Lie Groups

2011-01-01
Peter Schneider

MODULES OVER IWASAWA ALGEBRAS

2003-01-01
J. Coates , Peter Schneider , R. Sujatha
Let be a compact -valued -adic Lie group, and let be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion -modules, up to … Let be a compact -valued -adic Lie group, and let be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion -modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their -power torsion points.AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70
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Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures

2012-03-06
Florian Sprung
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Presentation of an Iwasawa algebra: the case of $\Gamma_1{SL}(2,\Bbb Z_p)$

2011-01-01
Laurent Clozel
We give an explicit presentation of the p -adic Iwasawa algebra of the subgroup of level one of SL(2, {Z}_{p}) for p \neq 2. We give an explicit presentation of the p -adic Iwasawa algebra of the subgroup of level one of SL(2, {Z}_{p}) for p \neq 2.
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Lectures on Chevalley Groups

2016-12-20
Robert Steinberg

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>

2018-06-30
Jishnu Ray

On the p-adic L-function of a modular form at a supersingular prime

2003-06-15
Robert Pollack
In this paper we study the two $p$-adic $L$-functions attached to a modular form $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, we are able to … In this paper we study the two $p$-adic $L$-functions attached to a modular form $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, we are able to decompose both the sum and the difference of the two unbounded distributions attached to $f$ into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of $f$ (and the fact that $a\sb p$ vanishes). From this description we explain how the $p$-adic $L$-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas for the $p$-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.

Sur les <i>p</i>‐extensions des corps <i>p</i>‐rationnels

1990-01-01
Par A. Movahhedi

Globally Analytic p-adic Representations of the Pro–p Iwahori Subgroup of GL(2) and Base Change, II: A Steinberg Tensor Product Theorem

2018-01-01
Laurent Clozel
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Locally analytic vectors in representations of locally 𝑝-adic analytic groups

2017-02-01
Matthew Emerton
This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any … This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any continuous representation of a p-adic analytic group on a locally convex p-adic topological vector space the associated space of locally analytic vectors. Using this functor, and the point of view that its construction suggests, we establish some basic facts about admissible locally analytic representations (as defined by Schneider and Teitelbaum). We also introduce the related notion of essentially admissible locally analytic representations.
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Iwasawa theory for elliptic curves

1999-01-01
Ralph Greenberg
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Explicit ring-theoretic presentation of Iwasawa algebras

2018-10-30
Jishnu Ray
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Cohomology of Number Fields

2008-01-01
Jürgen Neukirch , Alexander Schmidt , Kay Wingberg

Limites de représentations cristallines

2004-10-15
Laurent Berger
Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example -modules attached to crystalline representations, which allows us to … Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example -modules attached to crystalline representations, which allows us to improve some results of Fontaine, Wach and Colmez.
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Globally analytic $p$-adic representations of the pro--$p$--Iwahori subgroup of $GL(2)$ and base change, I : Iwasawa algebras and a base change map

2017-08-30
Laurent Clozel

Rational points of abelian varieties with values in towers of number fields

1972-12-01
Barry Mazur

Galois representations with open image

2016-02-05
Ralph Greenberg
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Banach space representations and Iwasawa theory

2002-12-01
P. Schneider , J. Teitelbaum
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EULER CHARACTERISTICS AS INVARIANTS OF IWASAWA MODULES

2002-10-14
Susan Howson
If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of … If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves. 2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

Euler characteristics and elliptic curves II

2001-01-01
John Coates , Susan Howson
This paper describes a generalisation of the methods of Iwasawa Theory to the field F∞ obtained by adjoining the field of definition of all the p-power torsion points on an … This paper describes a generalisation of the methods of Iwasawa Theory to the field F∞ obtained by adjoining the field of definition of all the p-power torsion points on an elliptic curve, E, to a number field, F. Everything considered is essentially well-known in the case E has complex multiplication, thus it is assumed throughout that E has no complex multiplication. Let G∞ denote the Galois group of F∞ over F. Then the main focus of this paper is on the study of the G∞-cohomology of the p∞- Selmer group of E over F∞, and the calculation of its Euler characteristic, where possible. The paper also describes proposed natural analogues to this situation of the classical Iwasawa λ-invariant and the condition of having μ-invariant equal to 0. The final section illustrates the general theory by a detailed discussion of the three elliptic curves of conductor 11, at the prime p=5.
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Fine Selmer groups of elliptic curves over p-adic Lie extensions

2005-01-12
J. Coates , R. Sujatha

Iwasawa Theory for $p$-adic Representations

2018-06-20
Ralph Greenberg
Greenbergfor certain constants av, av•• We write the functional equation in the above way to focus on the behavior of Lv(s) at s= 1, which seems convenient for formulating our … Greenbergfor certain constants av, av•• We write the functional equation in the above way to focus on the behavior of Lv(s) at s= 1, which seems convenient for formulating our conjectures and results.But replacing V by the Tate twist V(l-n)={Vz(l -n)} and noting that Lvo-nh)=Lv(s+n-1) will give appropriate statements about Lv(s) at s=n, for any integer n.Let rv denote the order of pole for I'v(s) at s=l.Since I'v(s) has no zeros, rv>O.Often rv is also the order of vanishing of Lv(s) at s= 1, but not always.(We mention two examples: (1) V ={Qz(l)}, Lv(s)=((s-1),Hasse-Weil L-function LE(s), rv=O, but Lv(l) can vanish.)Now, if cp is any even Dirichlet character, the twisted L-series Lv(s, cp) should also satisfy a functional equation similar to (7), with the same I'-factor, relating cp) to Lv.(s, cp-1 ).If we let cp vary over the characters of I'= Gal (Q00 /Q), regarded as Dirichlet characters, it seems reasonable to conjecture that Lv(s, cp) will have a zero of order exactly rv at s= 1, except possibly for finitely many cp.Sometimes this is easy to verify.A more subtle case is LE(s).Rohrlich [22] has proved the above conjecture in this case (i.e. Lv(l, cp)='s=O for all but finitely many cp e f) if Eis a Weil curve.Now our general philosophy is that the behavior of the L-functions Lv(s, cp) at s= 1 (cp e f) should somehow be reflected in the structure of the Selmer groups Svp;T/Qoo).Thus, the above remarks suggest the following conjecture.We assume p is ordinary for V and TP is any Ga• invariant lattice.Conjecture 1. Svp/Tp(Q00 ) has A-corank equal to rv.We will be able to prove the following weaker result by making use of Tate's calculation of Euler-Poincare characteristics and also the conjectural description of r v in terms of quantities attached to the representation space VP.We will have to also assume that Vis pure in the sense that it arises from a motive of pure weight.We believe the above conjecture even without this assumption, but its seems to be a more subtle question then.Theorem 1.If Vis pure, then corankA (Svp;T/Qoo))>rv, It is especially interesting to consider the case where rv=rv.=0.Then LvU) and Lv.(1) are critical values in the sense of Deligne [4].Deligne
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On Selmer Groups in the Supersingular Reduction Case

2020-10-13
Antonio Lei , R. Sujatha
Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study … Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.
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Locally analytic distributions and 𝑝-adic representation theory, with applications to 𝐺𝐿₂

2001-10-18
Peter Schneider , Jeremy Teitelbaum
In this paper we study continuous representations of locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analytic groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> … In this paper we study continuous representations of locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analytic groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in locally convex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector spaces, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {Q}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a spherically complete nonarchimedean extension field of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, along with interesting new objects such as the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on global sections of equivariant vector bundles on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic symmetric spaces. We introduce a restricted category of such representations that we call “strongly admissible” and we show that, when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application we prove the topological irreducibility of generic members of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic principal series for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 2 left-parenthesis double-struck upper Q Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_2(\mathbb {Q}_p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-valued representations of locally <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analytic groups.

$p$-adic Hodge theory and values of zeta functions of modular forms

2018-11-06
Kazuya Katô
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On the irreducibility of locally analytic principal series representations

2010-12-01
Sascha Orlik , Matthias Strauch
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {G}</mml:annotation></mml:semantics></mml:math></inline-formula>be a<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-adic connected reductive group with Lie algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation></mml:semantics></mml:math></inline-formula>. For … Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {G}</mml:annotation></mml:semantics></mml:math></inline-formula>be a<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-adic connected reductive group with Lie algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation></mml:semantics></mml:math></inline-formula>. For a parabolic subgroup<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P subset-of bold upper G"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mo>⊂<!-- ⊂ --></mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">G</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {P} \subset \mathbf {G}</mml:annotation></mml:semantics></mml:math></inline-formula>and a finite-dimensional locally analytic representation<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"><mml:semantics><mml:mi>V</mml:mi><mml:annotation encoding="application/x-tex">V</mml:annotation></mml:semantics></mml:math></inline-formula>of a Levi subgroup of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {P}</mml:annotation></mml:semantics></mml:math></inline-formula>, we study the induced locally analytic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {G}</mml:annotation></mml:semantics></mml:math></inline-formula>-representation<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W equals upper I n d Subscript bold upper P Superscript bold upper G Baseline left-parenthesis upper V right-parenthesis"><mml:semantics><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mi>Ind</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">P</mml:mi></mml:mrow></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">G</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">W = \operatorname {Ind}_{\mathbf {P}}^{\mathbf {G}}(V)</mml:annotation></mml:semantics></mml:math></inline-formula>. Our result is the following criterion concerning the topological irreducibility of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"><mml:semantics><mml:mi>W</mml:mi><mml:annotation encoding="application/x-tex">W</mml:annotation></mml:semantics></mml:math></inline-formula>: If the Verma module<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis German g right-parenthesis circled-times Subscript upper U left-parenthesis German p right-parenthesis Baseline upper V prime"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo>⊗<!-- ⊗ --></mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">p</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:msup><mml:mi>V</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">U(\mathfrak {g}) \otimes _{U(\mathfrak {p})} V’</mml:annotation></mml:semantics></mml:math></inline-formula>associated to the dual representation<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V prime"><mml:semantics><mml:msup><mml:mi>V</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">V’</mml:annotation></mml:semantics></mml:math></inline-formula>is irreducible, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"><mml:semantics><mml:mi>W</mml:mi><mml:annotation encoding="application/x-tex">W</mml:annotation></mml:semantics></mml:math></inline-formula>is topologically irreducible as well.
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Coleman maps and the<i>p</i>-adic regulator

2011-12-31
Antonio Lei , David Loeffler , Sarah Livia Zerbes
This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline … This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(\Gamma) the algebra of Qp-valued distributions on \Gamma = Gal(Qp(\mu (p^\infty) / Qp). Our first result determines the H(\Gamma)-elementary divisors of the quotient of D_{cris}(V) \otimes H(\Gamma) by the H(\Gamma)-submodule generated by (\phi * N(V))^{\psi = 0}, where N(V) is the Wach module of V. By comparing the determinant of this map with that of L_V (which can be computed via Perrin-Riou's explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato's main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions.
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Explicit presentation of an Iwasawa algebra: The case of pro-<i>p</i> Iwahori subgroup of SL<sub> <i>n</i> </sub>(ℤ<sub> <i>p</i> </sub>)

2019-11-08
Jishnu Ray
Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory … Abstract Iwasawa algebras of compact p -adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p -adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} which were uniform pro- p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {p&gt;n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro- p Iwahori subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro- p group.

Wach Modules and Iwasawa Theory for Modular Forms

2010-01-01
Antonio Lei , David Loeffler , Sarah Livia Zerbes
We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules.Let f = anq n be … We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules.Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular.By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p .Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve).Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group.This allows us to formulate a "main conjecture".Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.
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Über die Klassenzahlen algebraischer Zahlkörper

1950-06-01
Sigekatu Kuroda
Seit Herr Artin seine allgemeinen L -Funktionen, die mit Frobeniusschen Gruppencharakteren gebildet sind, entdeckt hat [1], sind die multiplikativen Relationen Dedekindscher ζ -Funktionen als additive Relationen zwischen den Frobeniusschen Gruppencharakteren … Seit Herr Artin seine allgemeinen L -Funktionen, die mit Frobeniusschen Gruppencharakteren gebildet sind, entdeckt hat [1], sind die multiplikativen Relationen Dedekindscher ζ -Funktionen als additive Relationen zwischen den Frobeniusschen Gruppencharakteren mit Erfolg untersucht worden. Durch diese Methode sind gewisse Klassenzahlrelationen in den folgenden Zeilen zu betrachten.
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Th�orie d'Iwasawa des repr�sentationsp-adiques sur un corps local

1994-12-01
Bernadette Perrin-Riou

Discrete logarithms and local units

1993-11-15
Oliver Schirokauer
Let K be a number field and (9 K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct … Let K be a number field and (9 K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct l e th powers in (9 K using smooth algebraic integers. This method makes use of approximations of the l -adic logarithm to identify l e th powers. One version we give is successful if the class number of K is not divisible by l and if the units in C K which are congruent to 1 modulo l e +1 are l e th powers. A second version only depends on Leopoldt’s conjecture. We use the technique of constructing l e th powers to find discrete logarithms in a finite field of prime order. Our method for computing discrete logarithms is closely modelled after Gordon’s adaptation of the number field sieve to this problem. We conjecture th at the expected running time of our algorithm is L p [1/3; (64/9) 1/3 + o(1)] for p-&gt; oo, where L p [ s; c ] = exp ( c (log q )s (log log q ) 1-8 ). This is the same running time as is conjectured for the number field sieve factoring algorithm.

Ring-theoretic properties of Iwasawa algebras: a survey

2006-01-01
Ken A. Brown , Konstantin Ardakov
This is a survey of the known properties of Iwasawa algebras, i.e., completed group rings of compact $p$-adic analytic groups with coefficients the ring $\Bbb Z\_p$ of $p$-adic integers or … This is a survey of the known properties of Iwasawa algebras, i.e., completed group rings of compact $p$-adic analytic groups with coefficients the ring $\Bbb Z\_p$ of $p$-adic integers or the field $\Bbb F\_p$ of $p$ elements. A number of open questions are also stated.
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On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions

2016-09-29
Antonio Lei , David Loeffler , Sarah Livia Zerbes
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In … We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
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Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres

1998-01-01
Stéphane Louboutin
Majorations explicites du r\'esidu au point 1 des fonctions z\^eta de certain $s$ corps de nombres By St\'ephane LOUBOUTIN ( Majorations explicites du r\'esidu au point 1 des fonctions z\^eta de certain $s$ corps de nombres By St\'ephane LOUBOUTIN (
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THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

2013-06-07
Byoung Du Kim
Abstract Suppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$ . We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ … Abstract Suppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$ . We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $ -extension has no proper $\Lambda $ -submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $ -extension. This work is analogous to Greenberg’s result in the ordinary reduction case.
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Heuristics on class groups

1984-01-01
Henri Cohen , H. W. Lenstra

The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

2005-06-01
John Coates , Takako Fukaya , Kazuya Katô +2 more
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Generic transfer for general spin groups

2006-03-01
Mahdi Asgari , Freydoon Shahidi
We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general … We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp4 to GL4 follows from our result as a special case
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Links between cyclotomic and ${GL}_2$ Iwasawa theory

2003-01-01
John Coates , Peter Schneider , R. Sujatha
We study, in the case of ordinary primes, some connections between the GL 2 and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication. We study, in the case of ordinary primes, some connections between the GL 2 and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication.
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Euler characteristics and elliptic curves

1997-10-14
John Coates , Susan Howson
Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F ∞ be the … Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F ∞ be the field obtained by adjoining to ℚ all p -power division points on E . Write G ∞ for the Galois group of F ∞ over ℚ. Assume that the complex L -series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G ∞ -Euler characteristic of the Selmer group of E over F ∞ . If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.
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The centre of completed group algebras of pro-$p$ groups.

2004-01-01
Konstantin Ardakov
We compute the centre of the completed group algebra of an arbitrary countably based pro- p group with coefficients in \mathbb Fp or \mathbb Zp . Some other results are … We compute the centre of the completed group algebra of an arbitrary countably based pro- p group with coefficients in \mathbb Fp or \mathbb Zp . Some other results are obtained.
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Kummer theory for abelian varieties over local fields

1996-01-18
J. Coates , Ralph Greenberg

On the structure of Selmer groups over 𝑝-adic Lie extensions

2002-03-18
Yoshihiro Ochi , Otmar Venjakob
The goal of this paper is to prove that the Pontryagin dual of the Selmer group over the trivializing extension of an elliptic curve without complex multiplication does not have … The goal of this paper is to prove that the Pontryagin dual of the Selmer group over the trivializing extension of an elliptic curve without complex multiplication does not have any nonzero pseudo-null submodule. The main point is to extend the definition of pseudo-null to modules over the completed group ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript p Baseline left-bracket left-bracket upper G right-bracket right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}_p[[G]]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of an arbitrary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic Lie group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> without <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torsion. For this purpose we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript p Baseline left-bracket left-bracket upper G right-bracket right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}_p[[G]]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an Auslander regular ring. For the proof we also extend some results of Jannsen’s homotopy theory of modules and study intensively higher Iwasawa adjoints.

L-Functions and Tamagawa Numbers of Motives

2007-01-01
Spencer Bloch , Kazuya Katô
The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. … The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values "modulo Q* multiples" in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.

Class fields of abelian extensions of Q

1984-06-01
Barry Mazur , Andrew Wiles