## Journal of Differential Geometry

- J. Differential Geom.
- Volume 109, Number 3 (2018), 379-409.

### The geometric torsion conjecture for abelian varieties with real multiplication

Benjamin Bakker and Jacob Tsimerman

#### Abstract

The geometric torsion conjecture asserts that the torsion part of the Mordell–Weil group of a family of abelian varieties over a complex quasi-projective 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 *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 $\overline{X}(1)$ parameterizing 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 $d$; the same is true for entire curves $\mathbb{C} \to \overline{X}_1 (\mathfrak{n})$. We also deduce some results about the birational geometry of Hilbert modular varieties.

#### Article information

**Source**

J. Differential Geom., Volume 109, Number 3 (2018), 379-409.

**Dates**

Received: 16 November 2015

First available in Project Euclid: 10 July 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1531188186

**Digital Object Identifier**

doi:10.4310/jdg/1531188186

**Mathematical Reviews number (MathSciNet)**

MR3825605

**Zentralblatt MATH identifier**

06917311

#### Citation

Bakker, Benjamin; Tsimerman, Jacob. The geometric torsion conjecture for abelian varieties with real multiplication. J. Differential Geom. 109 (2018), no. 3, 379--409. doi:10.4310/jdg/1531188186. https://projecteuclid.org/euclid.jdg/1531188186