The geometric torsion conjecture for abelian varieties with real multiplication

TL;DR

This paper proves the geometric torsion conjecture for abelian varieties with real multiplication, establishing uniform bounds on torsion related to the base curve's gonality using hyperbolic and algebraic geometry techniques.

Contribution

It confirms the conjecture for a specific class of abelian varieties and relates torsion bounds to gonality, employing novel hybrid geometric methods.

Findings

Torsion is uniformly bounded for abelian varieties with real multiplication.

Finitely many torsion covers contain d-gonal curves outside the boundary.

No entire curves map into the boundary for fixed torsion covers.

Abstract

The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we furthermore show that the torsion is bounded in terms of the $\mathit{gonality}$ of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert modular varieties $\overline X(1)$ parametrizing such abelian varieties. We show that only finitely many torsion covers $\overline X_1(\mathfrak{n})$ contain $d$-gonal curves outside of the boundary for any fixed $d$. We further show the same is true for entire curves $\mathbb{C}\rightarrow \overline X_1(\mathfrak{n})$.