2-Selmer groups of hyperelliptic curves with two marked points

TL;DR

This paper studies the average ranks of Jacobians of hyperelliptic curves with two marked points over Q, showing bounds on their 2-Selmer groups and the size of associated Selmer groups, advancing understanding of rational points.

Contribution

It establishes new bounds on the average Mordell-Weil rank and Selmer groups for a specific family of hyperelliptic curves with marked points.

Findings

Average Mordell-Weil rank bounded above by 5/2.

Average 2-Selmer group size bounded above by 6.

Average size of certain isogeny Selmer groups is exactly 2.

Abstract

We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also prove that the average size of the $\phi$-Selmer groups of a family of isogenies associated to this family is exactly 2.