## Advanced Studies in Pure Mathematics

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

Christopher Rasmussen

#### 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
Revised: 28 September 2011
First available in Project Euclid: 24 October 2018

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

Digital Object Identifier
doi:10.2969/aspm/06310449

Mathematical Reviews number (MathSciNet)
MR3051251

Zentralblatt MATH identifier
1321.14037

#### 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