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Arithmetic of abelian varieties with constrained torsion

Authors: Christopher Rasmussen and Akio Tamagawa
Journal: Trans. Amer. Math. Soc. 369 (2017), 2395-2424
MSC (2010): Primary 11G10; Secondary 11F80, 14K15
Published electronically: July 15, 2016
MathSciNet review: 3592515
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Abstract: Let $ K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $ K$ whose $ \ell $-power torsion fields are arithmetically constrained for some rational prime $ \ell $. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$ \ell $ fundamental group of $ \mathbb{P}^1-\{0,1,\infty \}$.

Under GRH, we demonstrate the set of classes is finite for any fixed $ K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of $ K/\mathbb{Q}$ and the dimension of abelian varieties are not too large through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of $ \ell $) are uniform in the degree of the extension $ K/\mathbb{Q}$.

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Additional Information

Christopher Rasmussen
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Akio Tamagawa
Affiliation: Research Institute for Mathematical Sciences, Kyoto 606-8502, Japan

Received by editor(s): October 30, 2013
Received by editor(s) in revised form: April 3, 2015
Published electronically: July 15, 2016
Additional Notes: The first author was partially supported by JSPS kakenhi Grant Number $19 ⋅07028$.
The second author was supported by JSPS kakenhi Grant Numbers 22340006, 15H03609.
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