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For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript p"> <mml:semantics> <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:annotation encoding="application/x-tex">\mathbb {Z}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-extension as one varies the prime <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> or the quadratic imaginary field in question.
For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb{Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.
For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb{Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.
Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, …
Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the ℤ p 2 -extension of a quadratic imaginary number field in which p splits in Section 6.
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.
N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. …
N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proportions and extend their results to the case of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$ belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.
Nous donnons une nouvelle description de la matrice de logarithme d'une forme modulaire en termes de distributions, g\'en\'eralisant le travail de Dion et Lei pour le cas $a_p=0$. Ce qui …
Nous donnons une nouvelle description de la matrice de logarithme d'une forme modulaire en termes de distributions, g\'en\'eralisant le travail de Dion et Lei pour le cas $a_p=0$. Ce qui nous permet d'inclure le cas $a_p\ne 0$ est une nouvelle d\'efinition, celle d'une matrice de distributions, et la caract\'erisation de cette matrice par de chiffres $p$-adiques. On peut appliquer ces m\'ethodes au cas correspondant d'une distribution \`a plusieurs variables. -- We give a new description of the logarithm matrix of a modular form in terms of distributions, generalizing the work of Dion and Lei for the case $a_p=0$. What allows us to include the case $a_p\ne 0$ is a new definition, that of a distribution matrix, and the characterization of this matrix by $p$-adic digits. One can apply these methods to the corresponding case of distributions in multiple variables.
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we …
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we …
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.
The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L ♯ p (f, T ) and L ♭ p (f, T ) …
The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L ♯ p (f, T ) and L ♭ p (f, T ) for a weight two modular form a n q n and a good prime p.This generalizes work of Pollack who worked in the supersingular case and also assumed a p = 0.The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.
Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(\Gamma_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which …
Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(\Gamma_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(\rho)=0$, then $\operatorname{Re}(\rho)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and "weighted moments" of critical values $L$-values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the $Z_f(s)$ keep track of arithmetic information. Assuming the Bloch--Kato Tamagawa Number Conjecture, they encode the arithmetic of a combinatorial arithmetic-geometric object which we call the "Bloch-Kato complex" for $f$. Loosely speaking, these are graded sums of weighted moments of orders of \v{S}afarevi\v{c}-Tate groups associated to the Tate twists of the modular motives.
In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the …
In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the Birch and Swinnerton-Dyer conjectures for analytic rank 0 or 1.
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) …
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L …
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L ♯ (E,T) and L ♭ (E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) …
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.
We study the asymptotic growth of the p-primary component of the Šafarevič–Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. As …
We study the asymptotic growth of the p-primary component of the Šafarevič–Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. As an application, we explain formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.
For a weight two modular form and a good prime $p$, we construct a vector of Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use this vector to put …
For a weight two modular form and a good prime $p$, we construct a vector of Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use this vector to put the $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer for ordinary [MTT] and supersingular [BPR] primes on one footing. Looking at $L_p^\sharp$ and $L_p^\flat$ individually leads to a stronger conjecture containing an extra zero phenomenon. We also give an explicit upper bound for the analytic rank in the cyclotomic direction and an asymptotic formula for the $p$-part of the analytic size of the Šafarevič-Tate group in terms of the Iwasawa invariants of $L_p^\sharp$ and $L_p^\flat$. A very puzzling phenomenon occurs in the corresponding formulas for modular forms. When $p$ is supersingular, we prove that the two classical $p$-adic $L$-functions ([AV75],[VI76]) have finitely many common zeros, as conjectured by Greenberg.
This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic …
This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic curve in cyclotomic $\Z_p$-extensions of number fields.
We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, …
We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$ with the good growth properties of the classical Pollack $p$-adic $L$-functions that in fact match them exactly when $a_p=0$ and $p$ is odd. We then generalize Kobayashi's methods to define two Selmer groups $\Sel^{\sharp}$ and $\Sel^{\flat}$ and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$. We then use results by Kato to prove a divisibility statement.
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.
The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L ♯ p (f, T ) and L ♭ p (f, T ) …
The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L ♯ p (f, T ) and L ♭ p (f, T ) for a weight two modular form a n q n and a good prime p.This generalizes work of Pollack who worked in the supersingular case and also assumed a p = 0.The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.
We review the main conjecture for an elliptic curve on Q having good supersingular reduction at p and give some consequences of it. Then we define notions of λ-invariant and …
We review the main conjecture for an elliptic curve on Q having good supersingular reduction at p and give some consequences of it. Then we define notions of λ-invariant and μ-invariant in this situation, generalizing a work of Kurihara and deduce the behaviour of the order of the Shafarevich-Tategroup up the cyclotomic Z p -extension. On examples, we give some arguments which, by combining theorems and numeral calculations, allow to calculate the order of the p-primary part of the Shafarevich-Tategroup in cases that are not yet known (nontrivial Shafarevich-Tate group, curves of rank greater than 1). Nous faisons Ie point sur la conjecture principale pour une courbe elliptique sur Q ayant bonne réduction supersingulière en p et en donnons quelques conséquences. Puis nous définissons la notion de λ invariant et de μ invariant dans cette situation, généralisant un travail de Kurihara et en déduisons la forme de I'ordre du groupe de Shafarevich-Tate le long de la Z p -extension cyclotomique. Par des exemples, nous donnons quelques arguments qui, en alliant théorèmes et calculs numériques, permettent de calculer I'ordre de la composante p-primaire du groupe de Shafarevich-Tate dans des cas non connus jusqu'à présent (groupe de Shafarevich-Tate non trivial, courbes de rang ≥ 1).
We study the asymptotic growth of the p-primary component of the Šafarevič–Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. As …
We study the asymptotic growth of the p-primary component of the Šafarevič–Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. As an application, we explain formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.
Abstract Let E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the …
Abstract Let E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ 2 p -extension of an imaginary quadratic field K when the prime p splits completely over K / ℚ , and prove that they enjoy a property analogous to Mazur's control theorem. Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±- p -adic L -functions of elliptic curves.
We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the characteristic …
We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the characteristic polynomial of Frobenius. For fixed p, the asymptotic running time for a curve of genus g over the field of p^n elements is O(g^{4+ε} n^{3+ε}).
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 …
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 Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), 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 Beilinson–Flach 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).
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.
This paper is about computational and theoretical questions regarding $p$-adic height pairings on elliptic curves over a global field $K$. The main stumbling block to computing them efficiently is in …
This paper is about computational and theoretical questions regarding $p$-adic height pairings on elliptic curves over a global field $K$. The main stumbling block to computing them efficiently is in calculating, for each of the completions $K\_v$ at the places $v$ of $K$ dividing $p$, a single quantity: the value of the $p$-adic modular form $\mathbf{E}\_2$ associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of $\mathbf{E}\_2$ of an elliptic curve. We also discuss the $p$-adic convergence rate of canonical expansions of the $p$-adic modular form $\mathbf{E}\_2$ on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that $\mathbf{E}\_2$ is log convergent.
We explain how to use results from Iwasawa theory to obtain information about $p$-parts of Tate-Shafarevich groups of specific elliptic curves over $\mathbb {Q}$. Our method provides a practical way …
We explain how to use results from Iwasawa theory to obtain information about $p$-parts of Tate-Shafarevich groups of specific elliptic curves over $\mathbb {Q}$. Our method provides a practical way to compute $\#\Sha (E/\mathbb {Q})(p)$ in many cases when traditional $p$-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is greater than 1. We apply our results along with a computer calculation to show that $\Sha (E/\mathbb {Q})[p]=0$ for the 1,534,422 pairs $(E,p)$ consisting of a non-CM elliptic curve $E$ over $\mathbb {Q}$ with conductor $\leq 30{,}000$, rank $\geq 2$, and good ordinary primes $p$ with $5 \leq p < 1000$ and surjective mod-$p$ representation.
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We show that there is a relation between the leading term at s=1 of an L-function of an elliptic curve defined over an number field and the term that follows.
We show that there is a relation between the leading term at s=1 of an L-function of an elliptic curve defined over an number field and the term that follows.
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our …
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case.
In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. …
In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch–Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over Q without complex multiplication.
Let F be a number field unramified at an odd prime p and $$F_\infty $$ be the $$\mathbf {Z}_p$$-cyclotomic extension of F. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, …
Let F be a number field unramified at an odd prime p and $$F_\infty $$ be the $$\mathbf {Z}_p$$-cyclotomic extension of F. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, Büyükboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary $${{\,\mathrm{Gal}\,}}(\overline{F}/F)$$-representations. In particular, their construction applies to abelian varieties defined over F with good supersingular reduction at primes of F dividing p. Assuming that these Selmer groups are cotorsion $$\mathbf {Z}_p[[{{\,\mathrm{Gal}\,}}(F_\infty /F)]]$$-modules, we show that they have no proper sub-$$\mathbf {Z}_p[[{{\,\mathrm{Gal}\,}}(F_\infty /F)]]$$-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler–Poincaré characteristic of these Selmer groups, we obtain an explicit formula on the size of the Bloch–Kato Selmer group attached to these representations. Furthermore, for two such representations that are isomorphic modulo p, we compare the Iwasawa-invariants of their signed Selmer groups.
We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. …
We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish.
When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the …
When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
We complete the proof that every elliptic curve over the rational numbers is modular.
We complete the proof that every elliptic curve over the rational numbers is modular.
The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the $L$-functions associated to modular forms in the anticyclotomic tower of conductor …
The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the $L$-functions associated to modular forms in the anticyclotomic tower of conductor $p\sp \infty$ over an imaginary quadratic field. While the author's previous work proved that such $L$-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is $+1$, the present work includes the case where the sign is $-1$. In that case, it is shown that the derivatives of the $L$-functions are generically nonzero at the center. It is also shown that when the sign is $+1$, the algebraic part of the central critical value is nonzero modulo $\ell$ for certain $\ell$. Applications are given to the mu-invariant of the $p$-adic $L$-functions of M. Bertolini and H. Darmon. The main ingredients in the proof are a theorem of M. Ratner, as in the author's previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.
Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have recently constructed two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, they showed …
Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have recently constructed two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, they showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this short note, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix.
Abstract We study Kato and Perrin-Riou's critical slope p -adic L -function attached to an ordinary modular form using the methods of A. Lei, D. Loeffler and S. L. Zerbes, …
Abstract We study Kato and Perrin-Riou's critical slope p -adic L -function attached to an ordinary modular form using the methods of A. Lei, D. Loeffler and S. L. Zerbes, Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (2010), 475–528. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p . We use this decomposition to prove results on the zeros of the p -adic L -function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens in “Overconvergent modular symbols and p -adic L -functions”, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 1–42.
We give a construction of an arithmetic p-adic L function of an elliptic curve with good reduction at p, with values in the Dieudonné module at p. We give the …
We give a construction of an arithmetic p-adic L function of an elliptic curve with good reduction at p, with values in the Dieudonné module at p. We give the link with Mazur-Swinnerton-Dyer functions, with the Beilinson-Kato elements, give a main conjecture". We calculate the dominant coefficients at integer points which are predicted by p-adic variants of the conjectures of Birch and Swinnerton-Dyer and Bloch-Kato.
The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the …
The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda--structure of the polynomial ring as Borger's descent data to and suggests its role in possible relation of $\Gamma_{\bold{R}}$--factor to real geometry over $\bold{F}_1$ (cf. also [CoCons2]).
Since the analytic functional equation holds for the ±-p-adic L-functions constructed in [7], the algebraic functional equation for the ±-Selmer groups is expected to hold as well.In this paper, we …
Since the analytic functional equation holds for the ±-p-adic L-functions constructed in [7], the algebraic functional equation for the ±-Selmer groups is expected to hold as well.In this paper, we show it following the ideas of [1] and [4].
The first chapter of this article contains an exposition of the work of Iwasawa and Mazur on the arithmetic of Abelian varieties over cyclotomic fields. The study of questions arising …
The first chapter of this article contains an exposition of the work of Iwasawa and Mazur on the arithmetic of Abelian varieties over cyclotomic fields. The study of questions arising here leads us in the second chapter to the use of the zeta-function apparatus, and the conjectures of Weil and Birch - Swinnerton-Dyer; this permits us to obtain conditional formulae for the order of the Tate-Shafarevich group.
Abstract We prove the μ -part of the main conjecture for modular forms along the anticyclotomic Z p -extension of a quadratic imaginary field. Our proof consists of first giving …
Abstract We prove the μ -part of the main conjecture for modular forms along the anticyclotomic Z p -extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ -invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ -invariant.