Arithmetic statistics of Prym surfaces

TL;DR

This paper studies Prym abelian surfaces over Q, showing average Selmer group sizes and providing evidence for Poonen-Rains heuristics in non-principally polarized families using advanced algebraic and geometric methods.

Contribution

It introduces a novel analysis of Prym surfaces with (1,2) polarization, establishing average Selmer group sizes and bounds, and applies diverse techniques including Lie algebra embeddings and orbit counting.

Findings

Average size of the Selmer group is 3.

Average size of the 2-Selmer group is bounded above by 5.

Provides evidence supporting Poonen-Rains heuristics for non-principally polarized abelian varieties.

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.