Selmer groups of some families of genus 3 curves and abelian surfaces

Abstract: Manjul Bhargava and Arul Shankar have determined the average size of the $n$-Selmer group of the family of all elliptic curves over $\mathbb{Q}$ ordered by height, for $n$ at most $5$. In this talk we will consider a family of nonhyperelliptic genus $3$ curves, and bound the average size of the $2$-Selmer group of their Jacobians. This implies that a majority of curves in this family have relatively few rational points. We also consider a family of abelian surfaces which are not principally polarized and obtain similar results. The proof is a combination of the theory of simple singularities, graded Lie algebras and orbit-counting techniques.

number theory

Audience: researchers in the discipline

POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

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