Ryotaro Sakamoto
In this paper, we prove that the dimension of the $p$-Selmer group for an elliptic curve is controlled by certain analytic quantities associated with modular symbols, which is conjectured by Kurihara.
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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 $\overline{f}$.
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.
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $χ$ of $K$, the $n$-Selmer groups of $E_1^χ$ and $E_2^χ$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogu…
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.
We study the variation of Iwasawa invariants of the anticyclotomic Selmer groups of congruent modular forms under the Heegner hypothesis. In particular, we show that even if the Selmer groups we study may have positive coranks, the mu-invariant vanishes for one modular form if and only if it vanishes for the other, and that their lambda-invariants are related by an explicit formula. This generalizes results of Greenberg-Vatsal for the cyclotomic extension, as well as results of Pollack-Weston and Castella-Kim-Longo for the anticyclotomic extension when the Selmer groups in question are cotorsi…
We study the Selmer group associated to a $p$-ordinary newform $f \in S_{2r}(Γ_0(N))$ over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K/\mathbb{Q}$. Under certain assumptions, we prove that this Selmer group has no proper $Λ$-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors.
For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the non-degeneracy of the $p$-adic height pairing, we proved that the structure of the Selmer group with respect to $p$-power torsion points is determined by some analytic elements $\tildeδ_{m}$ defined from modular symbols. In this paper, we do not assume the main conjecture nor the non-degeneracy of the $p$-adic height pairing, and study the structure of Selmer groups, …
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence t…
We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the $\mathbf{Z}_p$-extension being considered, resulting in the local cohomology groups that appear in the definition of the Selmer groups being significantly larger than they are in the cas…
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 $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
Fine Selmer groups of modular forms
研究模形式的细Selmer群与理想类群的Iwasawa不变量，比较其秩和μ不变量。
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.
We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their si…
Selmer stability in families of congruent Galois representations
本文研究同余模形式族中Selmer群的变化，探讨在固定素数p≥5下，通过Greenberg局部条件定义的Selmer群稳定性。证明在某些条件下，满足p-秩稳定性的模形式数量至少增长为X(logX)^{α-1}。
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 $K$ be an imaginary quadratic field where $p$ splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic $\mathbf{Z}_p$-extension of $K$, showing that their $μ$-invariants vanish. This generalizes and gives a new proof of a recent result of Matar on the vanishing of the $μ$-invariants of plus and minus signed Selmer groups for elliptic curves.
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.
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。
Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich-Tate groups of elliptic curves. In this paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a "Heegner hypothesis". More precisely, we show that the structure of Shafarevich-Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga-Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the ord…
We prove a level raising mod $\ell=2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond-Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.
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 study the variation of the dimension of the Bloch-Kato Selmer group of a p-adic Galois representation of a number field that varies in a refined family. We show that, if one restricts ourselves to representations that are, at every place dividing $p$, crystalline, non-critically refined, and with a fixed number of non-negative Hodge-Tate weights, then the dimension of the Selmer group varies essentially lower-semi-continuously. This allows to prove lower bounds for Selmer groups "by continuity", in particular to prove some predictions of the conjecture of Bloch-Kato for modular forms.
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
Deformations of reducible Galois representations with large Selmer $p$-rank
本文推广 Hamblen–Ramakrishna 形变提升技术，对任意 $p\geq5$ 构造与模形式相关的 Galois 表示，证明其 Bloch-Kato Selmer 群可具有任意大的 $p$-秩。
Ultra Kolyvagin systems and higher Fitting ideals of Iwasawa Selmer groups
本文通过发展等变超Kolyvagin系统理论，利用经典Selmer群的超积构造修补Selmer群外幂中的类集合，计算Iwasawa Selmer群的结构至伪同构并证明无有限子模，应用于椭圆曲线的精细Selmer群和模形式的Rankin-Selberg卷积的Bloch-Kato Selmer群。
2-Selmer companion modular forms
本文研究了2-Selmer伴随模形式，证明了在剩余Galois表示同构条件下剩余Greenberg 2-Selmer群的同构性，并给出了秩差有界时的逆命题，推广了Mazur-Rubin关于椭圆曲线的结果。
Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families
本文通过构造协调sheaf，研究了Coleman族的Rankin-Selberg卷积的Selmer群在临界范围内的界限问题。
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。。
Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions
通过放宽特定素数的秩0 Selmer结构，构造辅助Kolyvagin系统，描述Selmer群的Fitting理想，并用模符号刻画椭圆曲线在有限阿贝尔扩张上的Galois结构。
Refined conjectures on Fitting ideals of BDP Selmer groups
本文在温和假设下证明了Kurihara精细化Iwasawa理论意义下椭圆曲线BDP Selmer群的强主猜想，并确定了其初始Fitting理想的结构由$p$-adic模形式在CM点的取值决定。
Selmer groups of symmetric powers of ordinary modular Galois representations
本文研究 Hida 族 Galois 表示偶对称幂的分圆 Selmer 群，在 $n\leq4$ 时将其特征幂级数联系于同余理想，并用 $\mathcal{L}$-不变量描述平凡零点首项。
Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that…
On a parity result for the symmetric square of modular forms with congruent residual representations
本文证明了具有同余剩余Galois表示的两个模形式相关联的对称平方表示在分圆扩张上Selmer群λ-不变量的奇偶性结果，并处理了普通与非普通两种情况。
For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $χ: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $Σ\supset \{\mathfrak{p} \mid p\}$ consider the Bloch-Kato Selmer group $H:=H^1_Σ(F, χ^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overlineρ$ giving rise to lines in $H$ which are modular by some $ρ_f$ (also unramified outside $Σ$) satisfies $d \geq n:= \dim_{\mathbf{F}} H$. This was proved under the assu…
Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic…
Refined applications of Kato's Euler systems for modular forms
本文探讨了Kato欧拉系统在高权模形式中的精细应用，证明了Mazur-Tate猜想并给出岩泽主猜想的新解释，同时涉及对BSD猜想的应用。
Kolyvagin's conjecture for modular forms at non-ordinary primes
本文在非普通素数情形下，利用带符号Selmer群及Tamagawa数猜想p-部分的结果，从Selmer群秩为一的假设出发证明了模形式的Kolyvagin猜想完整版本。
Unbounded average Selmer ranks of elliptic curves in torsion families
本文研究了具有特定 torsion 子群的椭圆曲线的 p-塞勒尔群平均大小，证明在许多情况下该平均值是无界的。
Structure of fine Selmer groups over $\mathbb{Z}_p$-extensions
本文研究非分圆 $\mathbb{Z}_p$-扩张上阿贝尔簇精细 Selmer 群的挠性猜想，证明其与 Coates-Sujatha 伪零化猜想及 Mazur 增长数猜想相容，并将研究推广到椭圆模形式和 Hida 族情形。
On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements
本文显式构造了半稳定椭圆曲线的秩1本原Stark系统，证明了其与极小模性提升定理的等价性，并比传统几何Euler系统方法更精细地解释了Flach的zeta元素。
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。
Sums of rational cubes and the $3$-Selmer group
本研究得到j不变量为0的椭圆曲线三次扭族中3-塞尔默群的分布，改进了两有理立方和整数密度的上下界，通过强化解析核心、证明三线性大筛法解决了此前方法的挑战。
Norm relations for CM points on modular curves
本文研究了模曲线上的CM点的规范关系，通过欧拉系统方法探讨Selmer群结构，并为更高维推广提供基础。
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.…
Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian
本研究提出一种计算亏格2曲线雅可比簇2-塞尔默群上Cassels-Tate配对的方法，可改进2-下降法得到的有理点群秩上界，且已用于确定LMFDB中所有亏格2雅可比簇的秩。
In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication case. The key input of our approach is the use of modularity techniques to prove the vanishing of certain Selmer groups involved in the Chabauty-Kim method.
Indivisibility of Heegner cycles over Shimura curves and Selmer groups
本文证明 Shimura 曲线上 Heegner 循环与 theta 元素构成二分 Euler 系统，并由此推导高权模形式的 Gross-Zagier-Kolyvagin 逆定理，表明剩余 Selmer 群秩为一时 Heegner 循环像非零。
Congruences of modular forms and modularity of Tate-Shafarevich classes
本文在适当假设下证明了有理数上椭圆曲线的$p$-挠Tate-Shafarevich类在模曲线Jacobi商群中的可见性，验证了Jetchev-Stein猜想，其核心方法是利用Bertolini-Darmon二部Kolyvagin系统的非平凡性来推导Selmer群模$p$的基。
The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves
本文利用弱于 Iwasawa 主猜想的结论刻画任意秩椭圆曲线的 Selmer 群结构，并推出 Kato Kolyvagin 系统非平凡、BSD 结构精细化、秩上界及 Kurihara 猜想等结果。
We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.
Flach system of Jacquet-Langlands type and quaternionic period
本文研究了模形式在定四元数代数上 Jacquet-Langlands 转移后的伴随 Selmer 群，证明其长度由四元数模形式的 Petersson 范数赋值给出，并基于 Flach 的 Euler 系统论证提出了一种不依赖 Siegel 模单位的新构造方法。
On local characterizations of Hida families of Siegel modular forms
本文通过局部特征化方法，研究了具有双曲二重的模形式Hida家族，并证明了相关Selmer群的伪零性及R=TT定理。
Consider elliptic curves $ E=E_σ: y^2 = x (x+σp) (x+σq), $ where$ σ=\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(ϕ)}(E/{\mathbf{Q})}$, and $\ S^{(\hatϕ)}(E/{\mathbf{Q})} $ are explicitly determined, e.g., $\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $ $({\mathbf{Z}}/2{\mathbf{Z}})^2; $ $ ({\mathbf{Z}}/2{\mathbf{Z}})^3; $ or $ ({\mathbf{Z}}/2{\mathbf{Z}})^4 $ when $p\equiv 5; 1 $ or $3; $ or $ 7 ({\mathrm{mod}} 8)$ respectively. (2) When $p\equiv 5 (3, 5$ for $σ=-1) ({\mathrm{mod}} 8), $ it is proved that the Mordell-Weil group $ E({\mathbf{Q})}…
It is shown that the modular symbol of a cusp form of weight two has logarithmic growth.
Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
该研究针对同余数曲线，建立了Qin归一化表示缺陷奇偶性与Cassels配对Pfaffian的关系，推导了特定条件下的同余数相关同余式并推广了相关结论。
Iwasawa Theory of Elliptic Curves in Quadratic Twist Families
本文通过代数和分析方法研究二次twists家族中椭圆曲线的Iwasawa不变量变化，涉及半整数权模形式和Selmer群的研究。
We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic p-adic L-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p-adic L-functions to more general Shimura curves.
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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