## Advanced Studies in Pure Mathematics

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

### An abelian surface with constrained 3-power torsion

#### Abstract

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

**Dates**

Received: 31 March 2011

Revised: 28 September 2011

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1540417826

**Digital Object Identifier**

doi:10.2969/aspm/06310449

**Mathematical Reviews number (MathSciNet)**

MR3051251

**Zentralblatt MATH identifier**

1321.14037

**Subjects**

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

#### Citation

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. https://projecteuclid.org/euclid.aspm/1540417826