Tag - Selmer groups

Sam Mundy: Vanishing of Selmer Groups for Siegel Modular Forms

21st March, 2024

Let π be a cuspidal automorphic representation of Sp2n over ℚ which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch-Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch-Kato conjectures.

The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of non-classical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by χ.

Jordan Ellenberg: Stable Homology and the BKPLR Heuristics Over Function Fields

1st November, 2023

A basic question in arithmetic statistics is:  what does the Selmer group of a random abelian variety look like?  This question is governed by the Poonen-Rains heuristics, later generalized by Bhargava-Kane-Lenstra-Poonen-Rains, which predict, for instance, that the mod p Selmer group of an elliptic curve has size p+1 on average.  Results towards these heuristics have been very partial but have nonetheless enabled major progress in studying the distribution of ranks of abelian varieties.

Jef Laga: Arithmetic statistics and graded Lie algebras

17th March, 2022

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves.

Naomi Sweeting: Kolyvagin’s Conjecture and Higher Congruences of Modular Forms

22nd April, 2021

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The proof builds on work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome.

Sam Mundy: Eisenstein series, p-adic deformations, Galois representations, and the group G2

1st April, 2021

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.

Alexander Smith: Selmer groups and a Cassels-Tate pairing for finite Galois modules

25th February, 2021

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.

Richard Hatton: Heegner points and self-points on elliptic curves

21st September, 2020

In the arithmetic of elliptic curves, we are interested in the construction of points on an elliptic curve. In particular, it has been shown that we are able to bound certain Selmer groups using modular points, specifically the use of Heegner points by Kolyvagin and self points by Wuthrich. We will define these points and will show how they can be used to create the bounds and its generalisations.

Alex Smith: 2∞-Selmer groups, 2∞-class groups, and Goldfeld’s conjecture

14th September, 2017

Take E/ℚ to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of E have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to 2k-Selmer groups for any k>1. In addition, using the close analogy between 2k-Selmer groups and 2k+1-class groups, we will prove that the 2k+1-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all k>1.