# Explicit Methods for Abelian Varieties

### Overview

Abelian varieties are fundamental objects in algebraic geometry with a long, rich history of study. They are indispensable in number theory, and an important source of practical settings for cryptography. Although there are wide-ranging general structure theorems, efficient explicit computational tools required for applications are only available in the simplest cases.

Although many powerful theoretical advances have led to significant breakthroughs (eg. the Taniyama-Shimura-Weil Theorem and Fermat's Last Theorem, Weil descent and the elliptic curve discrete logarithm problem), efficient explicit methods for computing with abelian varieties in general are not known. For example, all abelian varieties by definition have a group law, but to date efficient methods to compute it are restricted to Jacobian varieties of algebraic curves, and most existing literature treats only the simplest cases of elliptic and hyperelliptic curves. Similarly, explicit methods to compute arithmetic data on abelian varieties, or even to tabulate interesting examples and tables of abelian varieties, only exist for relatively simple cases. Cremona's extensive tables of elliptic curves over the rationals of bounded conductor are a well-known resource that has proved to be valuable to many researchers, but there is very little data available for other types of abelian varieties. Even for the simplest case of algebraic curves, there is considerable interest in improving the state-of-the-art in these areas.

The CRG on Explicit Methods for Abelian Varieties represents an extensive network of regional, national, and international researchers focused on improving the state-of-the-art in this area. Building on existing relationships, we plan new partnerships and joint activities that will enable us to continue working together for years to come.

### Organizers

- Jeff Achter (Colorado State)
- Amir Akbary (Lethbridge)
- Mark Bauer (Calgary)
- Nils Bruin (Simon Fraser)
- Craig Costello (Microsoft Research)
- Laurent Imbert (CNRS, Montpellier, France)
- Michael Jacobson (Calgary)
- David Jao (Waterloo)
- Kumar Murty (Toronto)
- Andreas Stein (Oldenburg, Germany)
- Bianca Viray (Washington)

## Research Interests

- Properties of Galois representations and automorphic representations attached to Abelian varieties
- Prym varieties and intermediate Jacobians
- Constructing varieties with given endomorphism rings, zeta functions and related properties
- Rational points on moduli spaces
- Point counting on varieties over finite fields
- Efficient group arithmetic for Abelian varieties
- Cryptographic applications of Abelian varieties

### Planned Activities

**2015 Activities:**

- May 25-29: Explicit Methods for Abelian Varieties: Kick-off Workshop, at UCalgary
- December 4-7: 2015 CMS Winter Meeting, special session on "Representation Theory"

**2016 Activities:**

- June 13-16: GANITA: Geometry, Algebra, Number Theory, and their Information Technology Applications, at the Fields Institute (shared themes)
- June 16-18: Summer School in Explicit Methods for Abelian Varieties, at UCalgary

**2017 Activities:**

- June 5-9: PIMS Workshop on Computational Arithmetic Geometry at Simon Fraser University
- June 29 - July 12: Workshop on Arithmetic Geometry and Computer Algebra at the University of Oldenburg, Germany

**Ongoing Activities:**