Advanced Studies in Pure Mathematics

An abelian surface with constrained 3-power torsion

Christopher Rasmussen

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In my talk at the Galois Theoretic Arithmetic Geometry meeting, I described recent joint work with Akio Tamagawa on a finiteness conjecture regarding abelian varieties whose $\ell$-power torsion is constrained in a particular fashion. In the current article, we introduce the conjecture and provide some geometric motivation for the problem. We give some examples of the exceptional abelian varieties considered in the conjecture. Finally, we prove a new result—that the set $\mathscr{A}(\mathbb{Q},2,3)$ of $\mathbb{Q}$-isomorphism classes of dimension 2 abelian varieties with constrained 3-power torsion is non-empty, by demonstrating an explicit element of the set.

Article information

Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 449-456

Received: 31 March 2011
Revised: 28 September 2011
First available in Project Euclid: 24 October 2018

Permanent link to this document euclid.aspm/1540417826

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Secondary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11G32: Dessins d'enfants, Belyi theory


Rasmussen, Christopher. An abelian surface with constrained 3-power torsion. Galois–Teichmüller Theory and Arithmetic Geometry, 449--456, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310449.

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