Li Guo
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.
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The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ be elliptic curves over a number field $F$ with semistable reduction at all primes $v|p$ such that the $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ and $E_2[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_1$ and $E_2$. Leveraging these results, congruence relations for the truncated Euler charac…
Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$
本文研究了伴随 $L$-函数与 $\mathrm{GL}_n$ 上同余理想之间的关系，并利用此结果探讨了自守形式的同余与伴随 $L$-函数之间的联系，特别地，在 CM 和完全实数域中，给出了某些 Selmer 组大小的下界。
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 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.
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…
In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the $π$-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-$p$ extension of an admissible $p$-adic Lie extension and the structure of the dual fine Selmer group over the said admissible $p$-adic Lie extension.
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 $f$ 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…
The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $α\frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $α$ is depending on the conductor of the chosen elliptic curves an…
The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $ϕ_k \colon E_k \to E_{-27k}$. We compute the average size of the $ϕ_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in\mathbb Z$, is less than 1.21 and over $23\%$ (resp. $41\%$) of the cur…
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…
Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank
本文确定了面积为2mn的希罗三角形导出的椭圆曲线族的2-塞尔默群，明确其在二次扭转下的变化规律，推广了已有研究，构造出具有任意大2-塞尔默秩的希罗尼亚椭圆曲线显式族。
Klingen Eisenstein series congruences and modularity
本研究构造Klingen Eisenstein级数与Siegel尖点形式的模ℓ同余，证明相关Bloch-Kato塞尔默群非零性，推导通用形变环的R=dvr定理，并给出非循环塞尔默群与非主Eisenstein理想的条件。
Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer gro…
A canonical generator for congruence ideals of Hida families
本文通过Ohta配对构造标准伴随$p$-adic $L$-函数，生成Hida族的同余理想，并将其与原始伴随Selmer群的特征函数联系起来。
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 $\overline{f}$.
Mazur's growth number conjecture and congruences
本文研究虚二次域上 Mazur 关于 Selmer 秩增长的猜想在 Galois 表示 $p$-同余下的保持性，证明了对特定三元组及模形式 Greenberg Selmer 群的类似结果，并得出非反圆扩张中 Mordell-Weil 秩的有界性。
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.
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。
In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus η\oplus τ$: W(E/K,1\oplus η\oplus τ)=(-1)^{< 1\oplusη\oplus τ, X_{p}(E/L)>}, where $E$ is an elliptic curve over $K,$ $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree 2 of $Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $% ε_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction abov…
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
On the congruence ideal associated to $p$-adic families of Yoshida lifts
本文研究 Yoshida 提升的 p-adic 族同余，证明经典特化均为 Yoshida 提升，并建立同余理想整除 Selmer 群特征理想及伪循环性。
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…
In this article we study the 2-Selmer groups of number fields $F$ as well as some related groups, and present connections to the quadratic reciprocity law in $F$.
This paper contains some results regarding the Iwasawa module structure of Selmer groups of elliptic curves with complex multiplication.
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.
Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $μ$-invariant of the dual Selmer group is independent of the choice of the lattice.
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…
Class Groups and Selmer Groups in Special Families
本研究聚焦非阿贝尔数域，考察椭圆曲线中由3-同源诱导的Selmer群与理想类群3-部分之间的联系。
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。
We explicitly compute the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve using the Albanese-Albanese definition of the pairing given by Poonen and Stoll. This leads to a new proof that a pairing defined by Cassels on the 2-Selmer groups of elliptic curves agrees with the Cassels-Tate pairing.
We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4.
Structure of fine Selmer groups in abelian p-adic Lie extensions
本文研究abelian p-adic Lie扩张中椭圆曲线的fine Selmer群，给出其平凡性与伪零性的新结果，并阐明其与广义Greenberg猜想的关系。
We prove that when all elliptic curves over $\mathbb{Q}$ are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive proportion (in fact, at least one fifth) of all 2-Selmer elements of elliptic curves, when ordered by height, do not lift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elements in the associated Tate--Shafarevich groups.
We extend work of Swinnerton-Dyer on the density of the number of twists of a given elliptic curve that have 2-Selmer group of a particular rank.
Congruences of first syntomic cohomology groups
研究一阶syntomic上同调群的同余性质，通过比较其模p^n约简与附带Breuil-Kisin模和Nygaard滤子的模p^{2n}约简的同构性，揭示局部Bloch-Kato Selmer群的细化结构。
We study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. We show that this growth is determined by certain local cohomology groups and determine necessary and sufficient conditions for these groups to be trivial. Under certain hypotheses this allows us to give necessary and sufficient conditions for there to be growth in the full $p$-Selmer group in a degree $p$ Galois extension.
This paper is an extension of the work of Dummit and Voight on modeling the 2-Selmer group of number fields. We extend their model to $S_n$-number fields of even degree and develop heuristics on the difference in the 2-ranks between the class group and narrow class group.
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.
Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension to study the dual Selmer group over the $PGL(2)$ extension.
We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the $\ell^{\infty}$-Selmer groups for a given degree $\ell$ twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.
A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y^2=x^3+a(x-b)^2. The formula is in terms of the three-ranks of the quadratic number fields Q(\sqrt{a}) and Q(\sqrt{-3a}) and various aspects of the arithmetic of these number fields. In addition a duality theorem is used to relate the dimension of the Selmer group of the three-isogeny with the dimension of the Selmer group of its dual isogeny.
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 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.
The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.
In this paper and its sequel, we develop a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. Given a prime $\ell$ and a number field F not containing $μ_{2\ell}$, this method also shows that the $\ell^{\infty}$-class groups in the family of degree $\ell$ cyclic extensions of F have a distribution consistent with the Cohen-Le…
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。
For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a fine signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.
Arithmetic statistics for the Fine Selmer group in Iwasawa theory
本文研究了与椭圆曲线相关的Fine Selmer群的Iwasawa不变量的算术统计问题。
Cotorsion of anti-cyclotomic Selmer groups on average
本文研究椭圆曲线在改变素数p或二次虚域时，Selmer群在反分圆Z_p-扩张上呈现余挠性质的平均情况。
In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.
We study the functional equation for the multi-signed Selmer groups for non-ordinary motives whose Hodge-Tate weights are $0$ and $1$, defined by Büyükboduk and the first named author. This generalizes simultaneously Greenberg's result for ordinary motives and Kim's result for supersingular elliptic curves.
We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated $\mathbb{Z}_p$-module. The relationship between this conjecture and Iwasawa's classical $μ=0$ conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.
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.
In this paper, we study completely faithful torsion $\mathbb{Z}_p[[G]]$-modules with applications to the study of Selmer groups. Namely, if $G$ is a nonabelian group belonging to certain classes of polycyclic pro-$p$ group, we establish the abundance of faithful torsion $\mathbb{Z}_p[[G]]$-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such $\mathbb{Z}_p[[G]]$-modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer gr…
We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.
Iwasawa theory of fine Selmer groups over global fields
本文研究了全球数域上椭圆曲线的p^∞-细Selmer群的Iwasawa理论，探讨其与经典Iwasawa理论问题的联系，并提出与Jannsen猜想相关的研究
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…
Let A be an abelian variety defined over anumber field F. In this paper, we will investigate the growth of p-rank of the fine Selmer group in three situations. In particular, in each of these situations, we show that there is a strong analogy between the growth of the p-rank of the fine Selmer group and the growth of the p-rank of the class groups.
We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation a…
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