A comparison problem for abelian surfaces and descent for symplectic orbital integrals

TL;DR

This paper computes specific orbital integrals in the group GSp_4 related to the distribution of products of elliptic curves in isogeny classes of abelian surfaces over finite fields, advancing understanding of their arithmetic properties.

Contribution

It provides explicit calculations of orbital integrals for spherical functions over the subgroup GL_2×_{det} GL_2 and GSp_4, aiding in the analysis of abelian surface distributions.

Findings

Computed orbital integrals over the subgroup GL_2×_{det} GL_2

Calculated orbital integrals over GSp_4 in numerous cases

Contributed to understanding the distribution of elliptic curve products

Abstract

To answer a question about the distribution of products of elliptic curves in isogeny classes of abelian surfaces defined over finite fields, we compute specific orbital integrals in the group $\mathrm{GSp}_4$. More precisely, we compute integrals over the orbits of elements in the subgroup $\mathrm{GL}_2\times_{\det} \mathrm{GL}_2$. As a first step towards a complete solution of the problem, this article contains explicit computations for arbitrary orbital integrals of spherical functions over this subgroup, and also compute orbital integrals over $\mathrm{GSp}_4$ in a large number of cases.