The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point

Abstract

We prove that when all hyperelliptic curves of genus $n\ge 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\ge 3$ with a rational Weierstrass point have fewer than 20 rational points.

arXiv comment53 pages

References in corpus (2)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• Maximal linear spaces contained in the base loci of pencils of quadrics

Cited by in corpus (19)

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• Maximal linear spaces contained in the base loci of pencils of quadrics
• A positive proportion of Thue equations fail the integral Hasse principle
• A formula for certain Shalika germs of ramified unitary groups
• The average number of integral points on elliptic curves is bounded
• The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3
• Effective Chabauty for symmetric powers of curves
• Counting $S_5$-fields with a power saving error term
• On base sizes for algebraic groups
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• A positive proportion of locally soluble quartic Thue equations are globally insoluble
• Uniformity of fibres of period mappings and the $S$-unit equation
• Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields
• 2-Selmer groups of hyperelliptic curves with two marked points
• The second moment of the size of the $2$-Selmer group of elliptic curves