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arXiv · 2505.19285

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

Thomas Rüd
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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.

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Thomas Rüd. 2025-05-25. A comparison problem for abelian surfaces and descent for symplectic orbital integrals. https://arxiv.org/abs/2505.19285

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