UBC Number Theory Seminar 2026-2027

This talk is about Arthur packets and the Ramanujan-Petersson Conjecture. We will look at how recent progress on local Arthur packets leads to improved control on the non-temperedness of local components of automorphic representations.

Let \(E\) be an elliptic curve over an imaginary quadratic number field \(K\) and let \(p\) be an odd prime number. Greenberg introduced a topology for the family of linearly independent \(\mathbb{Z}_p\) extensions of \(F\) for any number field \(F\). A conjecture of Mazur predicts that the growth of rational points of \(E\) along a \(\mathbb{Z}_p\) extension of \(K\) is bounded unless the extension is the anticyclotomic extension, and the root number of \(E\) over K is \(-1\). We study the growth of rational points of \(E\) in the family of \(\mathbb{Z}_p\) extensions and its implications for Mazur's conjecture.

A classical method for exhibiting non-trivial elements in the Tate-Shafarevich group of Jacobians compares Selmer groups of various isogenies. In this presentation we consider Jacobians of plane quartics that have a (2,2,2)-isogeny to another principally polarized abelian variety. An interesting complication is that such an abelian variety may not be a Jacobian: it may only be a quadratic twist of a Jacobian. We describe how, at least in the case of completely split 2-torsion, one can determine which quadratic twist and how one can still compute Selmer groups. This talk describes joint work with Damara Gagnier and with Emiel Haakma.

Perfect codes in the Hamming metric over prime power alphabets were completely classified in the 1970s. For more general alphabets, analogous classifications were obtained for such codes correcting at least three errors. The case of 2-error correcting codes over non-prime power alphabets is apparently rather more difficult to handle. In this talk, I will discuss recent work attacking this problem via a variety of results from Diophantine approximation, both effective and ineffective.

I'll try to sketch some of the many answers proposed to the question in the title of the talk, starting with the most elementary attempts, and ending with some of the more sophisticated ones.

In 1956, Schäffer made a conjecture on the whole number solutions to the Diophantine equation \(S_k(x):=1^k+2^k+\cdots+x^k=y^n\). Towards this, we show how, for even \(k\), we can descend to a system of equations of the form \(Ax^p-By^p=1\), where \(A,B\in\mathbb{Q}\). In this presentation, we will introduce the necessary background to understand the modular method and then demonstrate how we can systematically determine all of the possible solutions to a system of equations of the form \(Ax^p-By^p=1\). We conclude by applying the techniques developed to Schäffer's conjecture and verify the conjecture for even \(k\le42\).

Local components of important classes of automorphic representations associated to a number field can be organised into sets of representations of p-adic groups - these sets are called local Arthur packets. Computation of Arthur packets is an important question in representation theory. They are characterized by Arthur's conjectures, which can be resolved using methods in harmonic analysis (endoscopic classification, theta correspondence) or algebraic geometry (vanishing cycles on perverse sheaves on certain moduli spaces of Langlands parameters). In this talk, I will give an introduction to local Arthur packets and report on my work in realizing them over p-adic groups.