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arXiv · 1307.3531

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

Arul Shankar·Xiaoheng Wang
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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.

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Arul Shankar, Xiaoheng Wang. 2014-02-17. Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves. https://doi.org/10.1112/s0010437x17007515

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