Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves

Abstract

In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mordell-Weil group of the Jacobians of these hyperelliptic curves. Finally using an equidistribution result, we modify the techniques of [16] to conclude that as $g$ tends to infinity, a proportion tending to 1 of these monic even-degree hyperelliptic curves having genus $g$ have exactly two rational points - the marked point at infinity and its hyperelliptic conjugate.

arXiv admin note: text overlap with arXiv:1208.1007 by other authors

References in corpus (5)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• Squarefree values of polynomial discriminants I
• Most odd degree hyperelliptic curves have only one rational point
• Maximal linear spaces contained in the base loci of pencils of quadrics
• Arithmetic invariant theory

Cited by in corpus (9)

• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3
• A threefold violating a local-to-global principle for rationality
• On the proportion of locally soluble superelliptic curves
• The second moment of the size of the $2$-Selmer group of elliptic curves
• Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields
• Fields generated by points on superelliptic curves
• Geometry-of-numbers methods in the cusp
• Monogenicity and 2-torsion in the class group of number fields of odd degree