We consider a family of abelian surfaces over 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 F4 ⊂ E6, invariant theory, a classical geometric construction due to Pantazis, a study of Néron component groups of Prym surfaces and Bhargava's orbit-counting techniques.

We consider a family of abelian surfaces over $$\mathbb {Q}$$ 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 $$F_4\subset E_6$$ F 4 ⊂ E 6 , invariant theory, a classical geometric construction due to Pantazis, a study of Néron component groups of Prym surfaces and Bhargava’s orbit-counting techniques.

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A simply laced Dynkin diagram gives rise to a family of curves over Q and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations.In this paper, we complete Thorne's picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang.The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck-Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.