## Journal of Differential Geometry

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

#### 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
First available in Project Euclid: 10 July 2018

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