Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields

We prove a version of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics for Selmer groups of quadratic twist families of abelian varieties over global function fields.

As a consequence, we derive a result towards the "minimalist conjecture" on Selmer ranks of abelian varieties in such families.

More precisely, we show that the probabilities predicted in these two conjectures are correct to within an error term in the size of the constant field, $q$, which goes to $0$ as $q$ grows.

Two key inputs are a new homological stability theorem for a generalized version of Hurwitz spaces parameterizing covers of punctured Riemann surfaces of arbitrary genus, and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.