On the parity of ranks of Selmer groups II

Controlling Selmer groups in the higher core rank case

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 …

Quadratic twists of abelian varieties and disparity in Selmer ranks

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree…

On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve

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.

Quadratic Twists of Elliptic Curves with Small Selmer Rank

Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial.

The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having…

Level raising mod 2 and arbitrary 2-Selmer ranks

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.

On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…

秩0的Kolyvagin系统与椭圆曲线在阿贝尔扩张上的Selmer群结构

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。

满足 Heegner 假设的二次扭转下的 Selmer 秩

Selmer ranks under quadratic twists satisfying the Heegner hypothesis

本文研究满足 Heegner 假设的二次扭转下 Selmer 秩的变化，推导出基于 $\mathbb{F}_2$ 矩阵的显式公式，并证明其与奇偶猜想相容。

具有任意大2-塞尔默秩的希罗尼亚椭圆曲线的二次扭转

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。

On 2-Selmer ranks of quadratic twists of elliptic curves

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.

Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…

Ranks of Selmer groups in an analytic family

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.

普通Rankin--Selberg特征簇上的Bessel周期与Selmer群

Bessel periods and Selmer groups over ordinary Rankin--Selberg eigenvariety

本文引入酉群及其Rankin--Selberg乘积的普通分布概念以重新定义普通特征簇，在定与不定情形下构造Bessel周期，并提出关联其消失除子与Selmer群特征除子的Iwasawa型猜想，证明了部分整除关系。

Selmer秩在偶Galois表示族中的密度，Wiles公式与全球互反律

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

本文通过Galois上同调方法研究偶Galois表示族中Selmer秩的分布，利用Wiles公式和全球互反律证明了Selmer秩增加的素数密度为1/192。

$\sqrt{-3}$-Selmer 群、理想类群与大的 $3$-Selmer 秩

$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

本文研究了具有有理 $3$-同源映射的椭圆曲线族 $E_{a,b}$，利用特定二次扩张的理想类群的 $3$-部分给出了其 $φ$-Selmer 群秩的上下界，并据此构造出无穷多条具有任意大 $3$-Selmer 秩且无阶为 $3$ 的非平凡有理点的曲线。

A Markov model for Selmer ranks in families of twists

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on a single parameter (the `disparity'), of a certain Markov process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolu…

Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles

Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.

CM阿贝尔簇扭变中的Selmer秩

Selmer ranks in twists of CM abelian varieties

本研究证明CM阿贝尔簇的若干p次扭变族中Selmer秩服从辛或酉分布，借此推出含本原p次单位根的数域上扭变费马曲线大多无解，并探讨了素数次循环扩张中的秩增长。

CM 阿贝尔簇的 p-Selmer 秩

p-Selmer ranks of CM abelian varieties

本文直接证明数域上复乘椭圆曲线的 p∞-Selmer 秩为偶数，并将该结果推广至阿贝尔簇。

Bounds for 2-Selmer ranks in terms of seminarrow class groups

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of…

Monsky矩阵与2-Selmer秩

Monsky Matrix and 2-Selmer rank

本文利用Monsky矩阵证明对应同余数椭圆曲线的2-Selmer秩为零，并建立量化结果，证明在模8余1,2,3的残差类中存在无限多个非同余数。

On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above $p$

Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $λ$-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other elliptic curve. In the midst of proving this latter result…

Finding large Selmer rank via an arithmetic theory of local constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is …

Root numbers and Selmer groups for the Jacobian varieties of Fermat curves

Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined an…

The growth of fine Selmer groups

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.

Disparity in Selmer ranks of quadratic twists of elliptic curves

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the…

Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202461$ curves with parameter he…

The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

We investigate in detail a homomorphism which we call the 2-Selmer signature map from the $2$-Selmer group of a number field $K$ to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields $K$ with given narrow class group 2-rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an appendix with Richard Foote, we classify the max…

The rank of 2-Selmer group associate to $θ$-congruent numbers

We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $π/3$ and $2π/3$-congruent numbers. Our second result gives some positive densities about $π/3$ and $2π/3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $π/3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $π/3$($resp.$ $2π/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.

Class number divisibility of $\mathbb{Q}(\sqrt{3p}, \sqrt{m-21pn^{2}})$ constructed from elliptic curves of $2$-Selmer rank exactly $1$

The class number divisibility problem for number fields is one of the classical problems in algebraic number theory, which originated from Gauss' class number conjectures. The relation between the points on an elliptic curve and class number divisibility of a number field has been explored through the works of various mathematicians. Here, we explicitly construct an unramified abelian extension of a bi-quadratic field generated from points of a certain type of elliptic curve. Moreover, showing the $2$-Selmer rank of the said elliptic curve as $1$, we also construct an infinite family of bi-qua…

Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves $E_{m, n}$ which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many ellip…

The average size of the 3-isogeny Selmer groups of elliptic curves $y^2 = x^3 + k$

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…

Multiplicities in Selmer groups and root numbers for Artin twists

Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the t…

有界平均塞勒尔秩的椭圆曲线在 torsion 家族中

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。

Elliptic curves with p-Selmer growth for all p

It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves E/Q whose 2-Selmer group goes up in every bi-quadratic extension and for any odd prime p, the p-Selmer group goes up in every D_{2p}-extension and every elementary abelian p-extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit …

具有大 Selmer p-秩的可约 Galois 表示的形变

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。

A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in this family, where r_2 is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.

p-adic families of Galois representations and higher rank Selmer groups

This is the final version of a book about p-adic families of Galois representations, Selmer groups, eigenvarieties and Arthur's conjectures

Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations

We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.

2-Selmer groups of hyperelliptic curves with two marked points

We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also prove that the average size of the $ϕ$-Selmer groups of a family of isogenies associated to this family is exactly 2.

关于具有同余剩余表示的模形式对称平方的一个奇偶性结果

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。

The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1

In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new l…

通过扭转变换增大 p-Selmer 秩

Increasing the p-Selmer rank by twisting

本文研究数域上椭圆曲线 p-扭转族的 p-Selmer 群，证明在特定同余与大小条件下，存在无穷多个特征使 p-Selmer 群维数达到任意给定值 d。

Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded.…

对Coleman族的Rankin-Selberg卷积的Selmer群进行界限研究

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。

$2$-Selmer 组的大小对于 $\frac{π}{3}$-congruent 数问题

The size of $2$-Selmer groups for the $\fracπ{3}$-congruent number problem

本文研究了 $\frac{π}{3}$-congruent 数问题中椭圆曲线的 $2$-Selmer 组的平均秩，并得出了相关密度结果。

关于 $\mathbb{Q}$ 上椭圆曲线二次扭的 2-Selmer 秩分布的刻画

On the Distribution of 2-Selmer ranks of Quadratic Twists of Elliptic Curves over $\mathbb{Q}$

本文提出带“洞”的随机交错矩阵模型，刻画了 $\mathbb{Q}$ 上椭圆曲线二次扭的 2-Selmer 秩分布，推广了先前结果，并发现同一族中不同等价类可具有不同分布。

The $\ell$-parity conjecture over the constant quadratic extension

For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…

The 2-Selmer group of $S_n$-number fields of even degree

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.

计算与模 p Galois 表示相关的 Selmer 组

Computing Selmer groups associated to mod p Galois representations

本文提出了一种计算与模 p Galois 表示相关的 Selmer 组的方法，并通过 Magma 实现，用于比较其秩与上同调类的周期，支持模 p Bloch-Kato 猜想的松散版本。

关于一族椭圆曲线的Mordell-Weil秩和2-Selmer群

On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves

本文研究了一族椭圆曲线的Mordell-Weil秩下界和2-Selmer群，扩展了之前关于参数族椭圆曲线Mordell-Weil秩的研究。

Governing fields and statistics for 4-Selmer groups and 8-class groups

Taking A to be an abelian variety with full 2-torsion over a number field k, we investigate how the 4-Selmer rank of the quadratic twist A^d changes with d. We show that this rank depends on the splitting behavior of the primes dividing d in a certain number field L/k. Assuming the grand Riemann hypothesis, we then prove that, given an elliptic curve E/Q with full rational 2-torsion, the quadratic twist family of E usually has the distribution of $4$-Selmer groups predicted by Delaunay's heuristic. Analogously, and still subject to the grand Riemann hypothesis, we prove that the set of quadrat…

Computing a Selmer group of a Jacobian using functions on the curve

In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a $(1-ζ_{p})$-Selmer group computation algorithm for the Jacobian of a curve of the form $y^{p}=f(x)$ where $p…