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

Thomas Rüd

TL;DR

The paper tackles the problem of understanding the distribution of products of elliptic curves within isogeny classes of abelian surfaces over finite fields by computing orbital integrals in $G=\mathrm{GSp}_4$ on the distinguished subgroup $H=\mathrm{GL}_2\times_{\det}\mathrm{GL}_2$. It develops a concrete, Kottwitz-style program that translates orbital integrals into point counts on Bruhat–Tits buildings, first for $\mathrm{SL}_2$ and then for the distinguished subgroup, yielding explicit Shalika germs for hyperbolic, elliptic, and mixed orbits. The work also demonstrates how non-elliptic cases can be handled by parabolic descent and initiates a symplectic analysis for equivalued elliptic elements using Moy–Prasad filtrations and affine Springer fibers to produce depth-dependent volume formulas. While not delivering Achter’s complete count, it provides a robust set of computational tools, explicit formulas, and a framework to advance the explicit determination of the isomorphism-class counts of abelian surfaces that decompose as products of elliptic curves. These methods pave the way for future exact computations across all orbit types and deepen connections between orbital integrals, Shimura varieties, and abelian-variety counting problems.

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.

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

TL;DR

The paper tackles the problem of understanding the distribution of products of elliptic curves within isogeny classes of abelian surfaces over finite fields by computing orbital integrals in on the distinguished subgroup . It develops a concrete, Kottwitz-style program that translates orbital integrals into point counts on Bruhat–Tits buildings, first for and then for the distinguished subgroup, yielding explicit Shalika germs for hyperbolic, elliptic, and mixed orbits. The work also demonstrates how non-elliptic cases can be handled by parabolic descent and initiates a symplectic analysis for equivalued elliptic elements using Moy–Prasad filtrations and affine Springer fibers to produce depth-dependent volume formulas. While not delivering Achter’s complete count, it provides a robust set of computational tools, explicit formulas, and a framework to advance the explicit determination of the isomorphism-class counts of abelian surfaces that decompose as products of elliptic curves. These methods pave the way for future exact computations across all orbit types and deepen connections between orbital integrals, Shimura varieties, and abelian-variety counting problems.

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

Paper Structure

This paper contains 46 sections, 70 theorems, 293 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A distinguished subgroup of the similitude group
  4. Statement of the problem.
  5. Relation to elliptic curves.
  6. Approaches to the problem.
  7. Kottwitz's approach and integrals over SL2.
  8. Orbital integrals as point count on the building
  9. Fixed points on the tree of SL2
  10. The case of SL2
  11. Setup
  12. Unipotent elements
  13. Regular semisimple elements: the hyperbolic case
  14. Regular semisimple elements: Unramified elliptic case
  15. Regular semisimple elements: Ramified elliptic case
  16. ...and 31 more sections

Key Result

Proposition 2.2

The group $\mathbf{G}(F)$ is not contained in any parabolic subgroup of $\mathrm{GSp}_{2n}(F)$.

Figures (17)

  • Figure 1: Building of $\mathrm{GL}_2(\mathbb{Q}_2)$ with coloring, preserved by the action of $\mathrm{SL}_2(\mathbb{Q}_2)$
  • Figure 2: Unramified quadratic extension of the building of $\mathrm{SL}_2$ over $\mathbb{Q}_2$
  • Figure 3: Ramified quadratic extension of the building of $\mathrm{SL}_2$ over $\mathbb{Q}_2$
  • Figure 4: Apartment of the building of $G$
  • Figure 5: Root system of $\mathrm{GSp}_4$ with choice of positive roots, the long roots is the root system of $\mathrm{GL}_2\times_{\det}\mathrm{GL}_2$. Shaded is the positive Weyl Chamber of $\mathrm{GSp}_4$. Below are root spaces
  • ...and 12 more figures

Theorems & Definitions (153)

  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.8: Langlands--Kottwitz formula, kottformula
  • Theorem 2.10: milnecount
  • Remark 2.11
  • Remark 3.1
  • ...and 143 more