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July 2018 The geometric torsion conjecture for abelian varieties with real multiplication
Benjamin Bakker, Jacob Tsimerman
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J. Differential Geom. 109(3): 379-409 (July 2018). DOI: 10.4310/jdg/1531188186

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.

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Benjamin Bakker. Jacob Tsimerman. "The geometric torsion conjecture for abelian varieties with real multiplication." J. Differential Geom. 109 (3) 379 - 409, July 2018. https://doi.org/10.4310/jdg/1531188186

Information

Received: 16 November 2015; Published: July 2018
First available in Project Euclid: 10 July 2018

zbMATH: 06917311
MathSciNet: MR3825605
Digital Object Identifier: 10.4310/jdg/1531188186

Rights: Copyright © 2018 Lehigh University

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Vol.109 • No. 3 • July 2018
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