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

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 GSp₄. More precisely, we compute integrals over the orbits of elements in the subgroup GL₂×_ GL₂. 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 GSp₄ in a large number of cases.