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arXiv · 2101.07658

Arithmetic statistics of Prym surfaces

Jef Laga
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Abstract

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of the Selmer group of this polarization equals $3$. Moreover we show that the average size of the $2$-Selmer group of the abelian surfaces in the same family is bounded above by $5$. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding $F_4\subset E_6$, invariant theory, a classical geometric construction due to Pantazis, a study of N\'eron component groups of Prym surfaces and Bhargava's orbit-counting techniques.

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Jef Laga. 2021-01-19. Arithmetic statistics of Prym surfaces. https://arxiv.org/abs/2101.07658

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