Abelian varieties not isogenous to Jacobians over global fields

TL;DR

The paper proves that over the function field $F=\overline{\mathbb{F}_p(t)}$ (and related settings over number fields and finite fields), there exist abelian varieties of dimension $g>1$ that are not isogenous to Jacobians. The authors develop a strategy based on hypersymmetric points and Serre–Tate coordinates to control $p$-power Hecke orbits, constructing curves with maximal monodromy that force any isogeny to lie outside a given divisor $D\subset \mathcal{A}_g$. They establish key local-to-global deformation phenomena via Serre–Tate directions, prove irreducibility of Hecke translates, and use intersection theory to guarantee the existence of points not isogenous to any point of $D$. The work yields a function-field result, a new number-field proof, and finite-field analogues, with broad implications for Torelli-type questions and the distribution of Jacobians among abelian varieties. The methods blend moduli-space geometry, $p$-adic deformation theory, and monodromy techniques to produce robust, quantitative counterexamples to isogeny-into-Jacobians across global fields.

Abstract

We prove the existence of abelian varieties not isogenous to Jacobians over characterstic $p$ function fields. Our methods involve studying the action of degree $p$ Hecke operators on hypersymmetric points, as well as their effect on the formal neighborhoods using Serre Tate co-ordinates. We moreover use our methods to provide another proof over number fields, as well as proving a version of this result over finite fields.

Theorems & Definitions (37)

• Definition 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Grothendieck, Raynaud
• proof
• Theorem 3.1
• Lemma 3.2
• proof