The Swinnerton-Dyer Seminar

Week 1 - 5th October 2026

Edison Au-YeungLink opens in a new window (University of Warwick) - The elliptic sieve and ideal norms

A theorem of Landau and Ramanujan states that almost all integers are not sums of two squares (when ordered by their absolute value). Equivalently, almost all integers are not norms from the Gaussian rationals $\mathbb{Q}(i)$. A version of this norm result for elliptic curves $E$ was proved by Bhakta et al. in 2023 using an argument known as the elliptic sieve. In this talk, I will first give an overview of the elliptic sieve, followed by a discussion of a version of the proof in which the norm result applies to abelian extensions of imaginary number fields.

Week 2 - 12th October 2026

Jiewei XiongLink opens in a new window (University of Reading) - A history of the Zilber–Pink conjecture

The talk doesn't contain any new mathematical content, rather it's a genealogical survey of the Zilber–Pink conjecture (both its original forms and subsequent variants) and preceding related problems, with some partial results (due to other people). I will use the motif of finiteness in diophantine geometry as a guide but if time allows I will also tell the story from the Hodge theoretic point of view.

Week 3 - 19th October 2026

Joseph Harrison - Effectivity in the semi-abelian Mordell conjecture

Let $G$ be a commutative algebraic group over $\mathbb{C}$, not containing a subgroup isomorphic to the additive group of complex numbers. Let $X$ be a closed subvariety of $G$ and $\Gamma$ a finitely generated subgroup of $G$. From the perspective of the group structure the set $X$ is unstructured, the notable exception being when $X$ is a translate of an algebraic subgroup. Similarly, $\Gamma$ is unstructured for the Zariski topology on $G$. A generalisation of the classical Mordell conjecture asks for a description of the Zariski closure of $X \cap \Gamma$. This was achieved for abelian varieties by Faltings and semi-abelian varieties by Vojta, but the proofs are ineffective. I will discuss an effective approach for certain $X$ and $\Gamma$ that does not make use of Diophantine approximation, linear forms in logarithms or the Chabauty—Skolem methods.