Arithmetic of abelian varieties with constrained torsion

2013/02/06 by Christopher Rasmussen, Rasmussen, Christopher, Akio Tamagawa +1 · 2 citations
Computer Science · Mathematics · #11G10 (Primary) 11F80 #14K15 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1302.1477

openalex publication_date 2013/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over K whose ℓ-power torsion fields are arithmetically constrained for some rational prime ℓ. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-ℓ fundamental group of P¹ - \0,1,∞\. 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/\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 ℓ) are uniform in the degree of the extension K/\Q.

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