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
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 ā¦
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 $p$.
We construct and study a triangulated category of motives with modulus that extends Voevodsky's category in such a way as to encompass non-homotopy invariant phenomena. It is intimately related to ā¦
We construct and study a triangulated category of motives with modulus that extends Voevodsky's category in such a way as to encompass non-homotopy invariant phenomena. It is intimately related to the theory of reciprocity sheaves, introduced by the authors in a previous paper. Contents
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant ā¦
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of \emph{proper modulus pairs}, that is, pairs of a proper $k$-variety $X$ and an effective divisor $D$ on $X$ such that $X \setminus |D|$ is smooth. To a modulus pair $(X, D)$ we associate its motive $M(X, D) \in \mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In some cases the Hom group in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.
For a motive M over Q , we define the fundamental periods of M using invariant theory. Our definition generalizes Deligne's periods. We show that if a motive M is ā¦
For a motive M over Q , we define the fundamental periods of M using invariant theory. Our definition generalizes Deligne's periods. We show that if a motive M is constructed from motives M 1 , M 2 ,..., M n by a standard algebraic operation, then the fundamental periods of M can be expressed as monomials of the fundamental periods of M 1 , M 2 ,..., M n . Applying this theory, we discuss two (hypothetical) motives attached to a Siegel modular form. We show that a Siegel modular form of degree m has at most m + 1 period invariants.
Let $E$ be an elliptic curve defined over a number field with good reduction at all primes above a fixed odd prime $p$, where at least one of which is ā¦
Let $E$ be an elliptic curve defined over a number field with good reduction at all primes above a fixed odd prime $p$, where at least one of which is a supersingular prime of $E$. In this paper, we will establish the algebraic functional equation for the mixed signed Selmer groups of $E$ over a multiple $\mathbb {Z}_{p}$-extension.
By Helmut Klingen: 162 pp., Ā£30.00, ISBN 0 521 35052 2 (Cambridge University Press, 1990).
By Helmut Klingen: 162 pp., Ā£30.00, ISBN 0 521 35052 2 (Cambridge University Press, 1990).
From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic ā¦
From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.
Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results ā¦
Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.
Abstract Let f be an elliptic modular form and p an odd prime that is coprime to the level of f . We study the link between divisors of the ā¦
Abstract Let f be an elliptic modular form and p an odd prime that is coprime to the level of f . We study the link between divisors of the characteristic ideal of the p -primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.
Let $V$ be a crystalline $p$-adic representation of the absolute Galois group $G_K$ of an finite unramified extension $K$ of $\mathbb {Q}_p$, and let $T$ be a lattice of $V$ ā¦
Let $V$ be a crystalline $p$-adic representation of the absolute Galois group $G_K$ of an finite unramified extension $K$ of $\mathbb {Q}_p$, and let $T$ be a lattice of $V$ stable by $G_K$. We prove the following result: Let $\mathrm {Fil}^1V$ be the maximal sub-representation of $V$ with Hodge-Tate weights strictly positive and $\mathrm {Fil}^1T=T\cap \mathrm {Fil}^1V$. Then, the projective limit of $H^1_g(K(\mu _{p^n}), T)$ is equal up to torsion to the projective limit of $H^1(K(\mu _{p^n}), \mathrm {Fil} ^1T)$. So its rank over the Iwasawa algebra is $[K:\mathbb {Q}_p]\operatorname {dim}\mathrm {Fil}^1 V$.
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 ā¦
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.
We study the Iwasawa Āµ-and Ī»-invariants of the non-primitive plus/minus Selmer groups of elliptic curves for supersingular primes.We prove that they are constant for a family of elliptic curves with ā¦
We study the Iwasawa Āµ-and Ī»-invariants of the non-primitive plus/minus Selmer groups of elliptic curves for supersingular primes.We prove that they are constant for a family of elliptic curves with the same residual representation if the Āµ-invariant of any of them is 0. As an application we find a family of elliptic curves whose plus/minus Selmer groups have arbitrarily large Ī»-invariants.
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 ā¦
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.
Abstract Let E be an elliptic curve over ā that has good supersingular reduction at p > 3. We construct what we call the Ā±/Ā±-Selmer groups of E over the ā¦
Abstract Let E be an elliptic curve over ā that has good supersingular reduction at p > 3. We construct what we call the Ā±/Ā±-Selmer groups of E over the ā¤ 2 p -extension of an imaginary quadratic field K when the prime p splits completely over K / ā , and prove that they enjoy a property analogous to Mazur's control theorem. Furthermore, we propose a conjectural connection between theĀ±/Ā±-Selmer groups and Loeffler's two-variable Ā±/Ā±- p -adic L -functions of elliptic curves.
Abstract This article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form ā¦
Abstract This article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define KobayashiāSprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) BeilinsonāFlach elements (out of the collection of unbounded BeilinsonāFlach elements of LoefflerāZerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral BeilinsonāFlach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they ā¦
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katzās $2$ -variable $p$ -adic $L$ -functions) and algebraic objects (two āeverywhere unramifiedā Iwasawa modules) involving codimension two cycles in a $2$ -variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$ , defined over $\mathbb{Q}$ , with good supersingular reduction at $p$ . On the analytic side, we consider eight pairs of $2$ -variable $p$ -adic $L$ -functions in this setup (four of the $2$ -variable $p$ -adic $L$ -functions have been constructed by Loeffler and a fifth $2$ -variable $p$ -adic $L$ -function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\mathbb{Z}_{p}^{2}$ -extension of $K$ . We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of CoatesāSujatha.
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 ā¦
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 ā Zp Ī, where Ī is the Iwasawa algebra of Z Ć 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 Ī-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.
Greenbergfor certain constants av, avā¢ā¢ 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 ā¦
Greenbergfor certain constants av, avā¢ā¢ 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 Gaā¢ 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