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On the pointwise convergence of the number of abelian varieties over $\mathbb{F}_p$ with fixed trace

Zhao Yu Ma, Jit Wu Yap, Jeff Achter, Julia Gordon

TL;DR

This paper proves the Ballini–Lombardo–Verzobio conjecture on the pointwise convergence of the distribution of the trace for genus $g$ abelian varieties over finite fields, extending Katz–Sarnak-type results to a product of local factors and the Sato–Tate measure. The authors develop a two-part strategy: (i) parametrize ordinary PPAVs by characteristic polynomials via Honda–Tate theory and establish a generic counting formula using Kottwitz’s orbital integrals, and (ii) sum these counts over all characteristic polynomials with fixed trace by employing an effective Chebotarev density theorem in families and an averaging argument for Euler products. They provide a detailed treatment of archimedean and non-archimedean local factors, including ratios of local zeta functions, stabilization phenomena for mod $l^k$ counts, and bounds for discriminants, all of which enable an $L^1$ convergence result and, in particular, consequences for genus $2$ and $3$ curves via the Torelli map. The work combines deep number-field techniques (Hilbert irreducibility, Chebotarev in families) with automorphic–geometric tools (Kottwitz’s framework, stable orbital integrals) to achieve explicit Euler-product expressions that converge to the predicted Sato–Tate-type limit. This advances our understanding of pointwise distributions in moduli spaces of abelian varieties and has direct implications for statistics of genus $2$ and $3$ curves over large finite fields. The results provide a robust framework for extending pointwise Katz–Sarnak-type refinements to higher-dimensional moduli, with explicit rates and a clear path to further generalizations.

Abstract

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p \to \infty$ for $\# A_g(\mathbb F_p,t)$, the number of $g$-dimensional PPAVs over $\mathbb F_p$ with trace $t$, as a product of natural local factors $v_\ell(t)$ for non-archimedean places $\ell$ and the Sato-Tate measure $\text{ST}_g$ corresponding to $\infty$. We prove that their conjecture is true for all $g$. As a consequence, we obtain analogous results on the distribution of curves of genus $2$ and $3$, answering questions of Bergström-Howe-García-Ritzenthaler [BHLR24] and [BLV25].

On the pointwise convergence of the number of abelian varieties over $\mathbb{F}_p$ with fixed trace

TL;DR

This paper proves the Ballini–Lombardo–Verzobio conjecture on the pointwise convergence of the distribution of the trace for genus abelian varieties over finite fields, extending Katz–Sarnak-type results to a product of local factors and the Sato–Tate measure. The authors develop a two-part strategy: (i) parametrize ordinary PPAVs by characteristic polynomials via Honda–Tate theory and establish a generic counting formula using Kottwitz’s orbital integrals, and (ii) sum these counts over all characteristic polynomials with fixed trace by employing an effective Chebotarev density theorem in families and an averaging argument for Euler products. They provide a detailed treatment of archimedean and non-archimedean local factors, including ratios of local zeta functions, stabilization phenomena for mod counts, and bounds for discriminants, all of which enable an convergence result and, in particular, consequences for genus and curves via the Torelli map. The work combines deep number-field techniques (Hilbert irreducibility, Chebotarev in families) with automorphic–geometric tools (Kottwitz’s framework, stable orbital integrals) to achieve explicit Euler-product expressions that converge to the predicted Sato–Tate-type limit. This advances our understanding of pointwise distributions in moduli spaces of abelian varieties and has direct implications for statistics of genus and curves over large finite fields. The results provide a robust framework for extending pointwise Katz–Sarnak-type refinements to higher-dimensional moduli, with explicit rates and a clear path to further generalizations.

Abstract

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as for , the number of -dimensional PPAVs over with trace , as a product of natural local factors for non-archimedean places and the Sato-Tate measure corresponding to . We prove that their conjecture is true for all . As a consequence, we obtain analogous results on the distribution of curves of genus and , answering questions of Bergström-Howe-García-Ritzenthaler [BHLR24] and [BLV25].
Paper Structure (48 sections, 65 theorems, 157 equations)

This paper contains 48 sections, 65 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Proof overview
  3. Acknowledgments
  4. Parametrization by characteristic polynomials
  5. Honda-Tate theory
  6. The region $\mathcal{R}_g$
  7. Irreducibility
  8. Generic Galois group
  9. Existence of PPAVs
  10. PPAV counts for a generic characteristic polynomial
  11. Kottwitz's orbital integral formula
  12. Reinterpretation as an Euler product
  13. Conjugacy classes in reductive groups
  14. Kottwitz triples and invariants
  15. Proof of Theorem \ref{['thm: PPAV counts for char poly']}
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $g\ge 2$ be an integer and let $\operatorname{USp}_{2g} = \operatorname{Sp}_{2g}(\mathbb{C}) \cap U(2g)$. Let the trace map $\mathop{\mathrm{tr}}\nolimits\colon \operatorname{USp}_{2g} \rightarrow [-2g,2g]$ send a matrix in the unitary symplectic group to its trace. Define the Sato-Tate measure converges in distribution as $p\rightarrow \infty$.

Theorems & Definitions (132)

  • Theorem 1.1: Katz-Sarnak for $\mathcal{M}_g$ KS99
  • Theorem 1.2: Katz-Sarnak for $\mathcal{A}_g$ KS99
  • Conjecture 1.4: $L^1$ Katz-Sarnak for $\mathcal{M}_g$ BLV
  • Conjecture 1.5: $L^1$ Katz-Sarnak for $\mathcal{A}_g$ BLV
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: $L^1$ Katz-Sarnak for $\mathcal{M}_2$ and $\mathcal{M}_3^{sym}$
  • Conjecture 1.9: Pointwise Global Katz-Sarnak for $\mathcal{M}_g$ and $\mathcal{A}_g$
  • Theorem 1.10
  • Definition 2.1
  • ...and 122 more