Yara Elias
We construct an Euler system of generalized Heegner cycles to bound the Selmer group associated to a modular form and an algebraic Hecke character. The main argument is based on Kolyvagin's machinery explained by Gross while the key object of the Euler system, the generalized Heegner cycles, were first considered by Bertolini, Darmon and Prasanna.
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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 $\overline{f}$.
We bound Selmer groups attached to Grossencharacters of CM elliptic curves by the appropriate $L$-value. Our method is to use Kato's explicit reciprocity law and the Main Conjecture as proved by Rubin. These results are then used together with work of Han and fundamental results of Nekovar to bound the torsion in the image of the $p$-adic cycle class map for self products of CM elliptic curves. Our results are valid for Chow groups of arbitrary codimension for almost all $p$.
Euler characteristic of the Selmer group attached to an Artin representation
本文遵循Greenberg与Vatsal的框架，将Longo和Vigni推广Greenberg所得的p进伽罗瓦表示相关塞尔姆群基数关系，拓展至阿廷表示的情形。
In this paper we study when two congruent $l$-adic Galois representations have congruent Selmer groups. We obtain results for representations from cyclotomic characters, Hecke characters and adjoints of modular forms.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $Λ$ is isomorphic to the ring of formal power series. By using the "Kolyvagin system" arising from a given Euler system $c$, we shall construct ideals $C_i(c)$ of $Λ$, and prove that the ideals $C_i(c)$ approximate the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we shall prove that the ideals $C_i(c)$ arising from the Euler system of Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual fine S…
We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.
We study the adjoint Bloch--Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\mathrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.
The main conjecture of Iwasawa theory is a conjecture on the relation between a Selmer group and a conjectural $p$-adic $L$-function. This conjectural $p$-adic $L$-function is expected to satisfy a conjectural functional equation in a certain sense. In view of the main conjecture and this conjectural functional equation, one would expect to have certain algebraic relationship between the Selmer group attached to a Galois representation and the Selmer group attached to the Tate twist of the dual of the Galois representation. It is precisely a component of this algebraic relationship that this p…
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $Γ_0(4).$
In this paper, we show that an action on the set of elliptic curves with j= 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $Γ_0(N)$ with $p\nmid N$.
Let $E$ be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over $\mathbb{Q}$ constructed from $E$ through multiplicative induction; it is of rank $8$. We show that, under certain assumptions on $E$, the non-vanishing of the central critical value of the $L$-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is $0$.
We prove that the "naive" convolution Dirichlet series D_2(s) attached to a degree 2 Siegel Hecke cusp form F, has a pole at s=1. As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor zeta function attached to F is s = 1.
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
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 generalization 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.
This paper answers a question of Gross and others, by exhibiting specific examples of Hecke algebras where mod 2 multiplicity one fails for some modular forms, and the associated Hecke algebras are not Gorenstein. It shows that the methods of Wiles, Gross, Edixhoven, Buzzard, Mazur and others have an upper bound to their applicability.
In this paper, we study the fine Selmer groups attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic p. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic $\Zp$-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups.
We define Kolyvagin systems and Stark systems attached to $p$-adic representations in the case of arbitrary `core rank' (the core rank is a measure of the generic Selmer rank in a family of Selmer groups). Previous work dealt only with the case of core rank one, where the Kolyvagin and Stark systems are collections of cohomology classes. For general core rank, they are collections of elements of exterior powers of cohomology groups. We show under mild hypotheses that for general core rank these systems still control the size and structure of Selmer groups, and that the module of all Kolyvagin …
Adjoint Selmer groups of automorphic Galois representations of unitary type
证明了酉类型自守Galois表示的伴随Selmer群消失，并为非CM Hilbert模形式和全实域椭圆曲线提供确切结果。。
Duality of Selmer groups of an abelian variety over a number field
本文研究数域上阿贝尔簇的Selmer群，通过奇素数$p$下的两个Selmer群及其对偶，建立其阶与对偶阿贝尔簇分支群阶之间的关系。
We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F . We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic Zp-extension and certain correction terms. This extends our earlier work [arXiv:math/0404431], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
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 sh…
Let $F$ be the element $\sum_{n\ \mathit{odd},\ n>0}x^{n^{2}}$ of $Z/2[[x]]$. Set $G=F(x^{5})$, $D=F(x)+F(x^{25})$. For $k>0$, $(k,10)=1$, define $D_{k}$ as follows. $D_{1}=D$, $D_{3}=D^{8}/G$, $D_{7}=D^{2}G$, $D_{9}=D^{4}G$; furthermore $D_{k+10}=G^{2}D_{k}$. Using modular forms of level $Γ_{0}(5)$ we show that the space $W$ spanned by the $D_{k}$ is stabilized by the formal Hecke operators $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p\ne 2$ or $5$. And we determine the structure of the (completed) shallow Hecke algebra attached to $W$. This algebra proves to be a power series ring in $T_{3}$ and …
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be another elliptic curve over a real quadratic number field. We construct a $\mathbb{Q}$-motive of rank $8$, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch-Kato Selmer group of the motive is $0$; and the non-vani…
Massey products and the Iwasawa theory of fine Selmer groups
本文用Massey积重新表述Coates-Sujatha猜想，证明μ-不变量的消失可传递至邻域扩张，并建立形变环无阻碍性与精细Selmer群μ-不变量消失的等价关系。
On the cohomology of GL(N) and adjoint Selmer groups
本文在局部-整体相容等条件下将Galatius-Venkatesh定理推广到CM域上GL(N)，构造从导出形变环基本群的可除群到扭对偶伴随动机Selmer群的Hecke等变单射，并推广到高阶同伦群与Hida族。
Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a…
Let $D$ in $Z/2[[x]]$ be $\sum x^{n^{2}}$, $n>0$ and prime to $6$. Let $W$ be spanned by the $D^{k}$, $k>0$ and prime to $6$. Then the formal Hecke operators $T_{p}$, $p>3$, stabilize $W$, and it can be shown that they act locally nilpotently. We show that the completion of the Hecke algebra generated by these $T_{p}$ acting on $W$, with respect to the maximal ideal generated by the $T_{p}$, is a power series ring in $T_{7}$ and $T_{13}$ with an element of square $0$ adjoined. This may be viewed as a level 3 analog of the level 1 results of Nicolas and Serre -- the Hecke stable space they stud…
An algorithm to compute Selmer groups via resolutions by permutations modules
针对数域、有限伽罗瓦模及塞尔默系统，提出通过赫克算子态射的分解算法计算塞尔默群Sel$_{\mathcal{L}}$，构造并证明$H^1_S(\mathcal{G}, M)$是包含Sel$_{\mathcal{L}}$的塞尔默群，还讨论了方法的时间复杂度。
On characteristic power series of dual signed Selmer groups
本文证明非寻常素数处模形式 Bloch-Kato Selmer 群的 p 主分支基数可由分圆 Z_p 扩张上有符号 Selmer 群特征幂级数常数项刻画，推广了寻常情形结果。
We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $Γ_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical line. This result extends the work of J.L. Hafner of 1983 where the case of the full modular group is considered.
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into probl…
In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible p-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain similar results for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.
In this article, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above $p$ to satisfy the $\M_H(G)$-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the $\M_H(G)$-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series.…
In 1992, Avner Ash and Mark McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a tar…
Refined conjectures on Fitting ideals of Selmer groups over $\mathbf{Z}_p^2$-extensions
本文在虚二次域 $K$ 的 $\mathbf{Z}_p^2$-扩张中，证明 Mazur-Tate 元素生成椭圆曲线对偶 Selmer 群的 Fitting 理想，从而精化相关猜想。
We investigate the existence of large values of L-functions attached to Maass forms on the critical line with prescribed argument. The results obtained rely on the resonance method developed by Soundararajan and furthered by Hough.
Let $f$ and $g$, of weights $k'>k\geq 2$, be normalised newforms for $Γ_0(N)$, for square-free $N>1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $λ\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,\frac{k+k'-2}{2})$. Under certain hypotheses, we prove that $λ$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we a…
In this paper, we study the Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the $\M_H(G)$-property, so will the other. In the event that the $\M_H(G)$-property holds, and assuming certain further hypothesis on the decomposition of primes in the $p$-adic Lie extension, we compare the ranks of the $π$-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also s…
In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical i…
The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi's definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above $p$. We then build on…
On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions
本文将分圆情形下权2赫克本征尖点形式的p准素塞尔默群与p进L函数的岩泽不变量关系，推广到反分圆情形，研究虚二次域下满足广义Heegner假设的两个形式的BDP塞尔默群与BDP p进L函数的岩泽不变量关联。
Factorization of Algebraic $p$-adic $L$-functions Attached to Adjoint Representations of Coleman Families: Non-critical Case
该研究针对模形式Coleman族伴随表示的代数$p$-进$L$-函数，借助多类数论工具证明含Selmer复形行列式的分解公式，建立$p$-进Artin形式体系并推广了前人结果至$p$-非平凡情形。
A formula of Perrin-Riou and characteristic power series of signed Selmer groups
证明 Kundu--Ray 猜想，给出超奇异约化椭圆曲线符号 Selmer 群特征幂级数首项的显式公式，基于 Perrin-Riou 的 p-adic L-函数公式。
Iwasawa cohomology of pseudorigid families of $(φ,Γ)$-modules
该研究证明伪刚性(φ,Γ)-模族相关岩泽上同调复形的性质，应用其给出Herr复形完美性的另一证明，还证明Selmer复形完美性并得到Selmer群秩变化的结果。
In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2,4 and 6 and their transformation under multiplication of the argument by $d$, and in particular, it determines uniquely the modular curve of degree $d$ isogenies between elliptic curves.
We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.
In this paper we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a structure analogous to the case of the Siegel modular group and coincides with the tensor product of its $p$-components for inert primes $p$. This leads to a characterization of the associated Siegel-Eisenstein series. The proof also involves Hecke theory for particular congruence subgroups.
We show how there is a natural action on the cohomology groups attached to certain subgroups of GL_n(F) of the Hecke operators defined as elements in an adelic double coset algebra. Our main result is, that if a system of eigenvalues for Hecke operators occur in the cohomology groups with coefficients in certain modules, then by changing the groups the system also occur in the cohomology groups with coefficients in a 1-dimensional module.
We construct an Euler system attached to a weight 2 modular form twisted by a Groessencharacter of an imaginary quadratic field, and apply this to bounding Selmer groups.
In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous…
Effective generation of Hecke algebras and explicit estimates of Sato--Tate type
本文在相关 L 函数满足 Riemann 假设的前提下，给出显式有限 Hecke 算子集，用以生成高权、无平方因子水平模形式空间上的 Hecke 代数。
The rationality of Stark-Heegner cycles attached to Bianchi modular forms -- The base-change scenario
本文证明了当Bianchi本征形式为经典椭圆尖点形式的基变换时，关于其Stark-Heegner循环为全局Bloch-Kato Selmer类限制的猜想成立。
On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers
本文研究模形式在反 cyclotomic Z_p 塔中的 Bloch-Kato 柯西群结构，证明在广义 Heegner 假设下，柯西群的 Pontryagin 对偶为自由模，并使 Bloch-Kato-Shafarevich-Tate 群消失。
Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional equation and mean square formula of $L_f(s)$. In this paper, we shall generalize these formulas for the derivatives of $L_f(s)$.
Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.
Variation of Iwasawa Invariants for Ordinary Representations
本文扩展Greenberg的岩泽不变量研究框架，比较普通p进表示的Selmer群在Greenberg邻域内ℤₚ扩张上的岩泽不变量，为精细Selmer群建立类似结果，并为分圆ℤₚ扩张邻域内相关特征理想与p进L函数的联系提供证据。
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