2-Selmer groups of hyperelliptic curves with two marked points

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 $\varphi$-Selmer groups of a family of isogenies associated to this family is exactly 2.

References in corpus (2)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• Maximal linear spaces contained in the base loci of pencils of quadrics