Powers of abelian varieties over $\overline{\mathbb Q(t)}$ not isogenous to a Jacobian

TL;DR

This work proves the existence of abelian varieties over $\overline{\mathbb{Q}(t)}$ whose powers are not isogenous to Jacobians, notably for dimensions $g=2^d$ with $d\ge4$, by leveraging Shimura curves of Mumford type and the Lu–Zuo Arakelov inequality. The main strategy combines precise control of Hodge line bundles, degree arguments on moduli spaces, and generalized Hecke correspondences to rule out any Jacobian isogeny for powers up to a prescribed bound, and then transfers these special-curve properties to generic curves over $\overline{\mathbb{Q}(t)}$ via intersection theory on suitable surfaces. The results yield two core contributions: (i) the explicit construction of abelian varieties over $\overline{\mathbb{Q}(t)}$ with no power isogenous to a Jacobian for $g\ge16$, and (ii) for any fixed $N$, the existence of abelian varieties with maximal monodromy such that no power up to $N$ is Jacobian-isogenous, thereby illustrating the singular behavior of Jacobian-power closures in arithmetic settings. The methods highlight a synergy between Arakelov bounds, moduli-theoretic correspondences, and monodromy-driven endomorphism rigidity to obstruct Torelli-type Jacobian realizations and advance understanding of the Torelli locus in arithmetic geometry.

Abstract

We prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with no power isogenous to a Jacobian. Moreover, given a positive integer $N$, we prove the existence of abelian varieties over $\overline{\mathbb Q(t)}$ with maximal monodromy such that the $n$th power is not isogenous to a Jacobian for $n \leq N$. We make use of an Arakelov inequality established by Lu and Zuo, as well as intersection theoretic methods, to prove our main results.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Remark 3.2
• Definition 3.3
• Remark 3.4
• Lemma 3.5
• proof
• Lemma 4.1