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The effective Shafarevich conjecture for abelian varieties of ${\text {GL}_{2}}$-type

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 May 2021

Rafael von Känel
Rafael von Känel*
Affiliation:
IAS Tsinghua University, Beijing 100084, China; E-mail:rafaelvonkanel@gmail.com.

Abstract

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In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$-type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 11G50: Heights

Information

Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e39
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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