Rational points and Selmer groups of genus 3 curves

Abstract: Manjul Bhargava and Arul Shankar have determined the average size of the n-Selmer group of the family of all elliptic curves over Q ordered by height, for n at most 5. They used this to show that the average rank of elliptic curves is less than one.

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.

number theory

Audience: researchers in the topic

London number theory seminar

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