Preface

This volume is the record of the special session on computational aspects of

arithmetic geometry at the April, 2006 regional meeting of the American Mathe-

matical Society at San Francisco State University. Our session attracted a large

number of outstanding researchers in the field, many of whom presented talks at

the session and contributed manuscripts to this volume.

In planning our gathering, we decided to stress computational issues because

of recent and ongoing increases in computing power. These increases have enabled

computational breakthroughs in any number of areas of arithmetic geometry. For

example, Cremona's tables of elliptic curves now go up to conductor 120,000 instead

of just conductor 1,000; tables of Hilbert class fields are known for discriminant up

to at least 5,000; and special values of Hilbert and Siegel modular forms can be

calculated to extremely high precision. In many cases, these experimental capabil-

ities have led to new observations and ideas for progress in the field. They have

also led to natural algorithmic questions on the feasibility and efficiency of many

computations, especially for the purpose of applications in cryptography.

As our gathering took shape, we were able to identify themes that united clus-

ters of talks: modular abelian varieties; Tate-Shafarevich groups of elliptic curves;

Stark's conjectures; Hilbert modular varieties; curves with complex multiplication.

In scheduling the lectures, we grouped these clusters together whenever possible.

A thread that ran through a number of the talks was the application of num-

ber theory and algebraic geometry to cryptography and coding theory. Potential

applications in these areas have suggested interesting mathematical questions and

computational challenges that are addressed by articles in this volume. For exam-

ple, the article by Bach and Charles addresses hardness of computing eigenforms

and relates it to breaking RSA cryptosystems based on the hardness of factoring

integers. The contribution by Broker and Stevenhagen addresses the issue of gen-

erating elliptic curves for use in cryptography by finding an efficient algorithm for

generating good parameters for using the CM (complex multiplication) method.

The contribution by Deajim and Grant uncovers interesting number theory ques-

tions when constructing good space-time codes. Two papers relate to zeta functions

of curves and surfaces, one providing algorithms for computing zeta functions and

one answering the question of which zeta functions of supersingular genus 2 curves

are possible over finite fields of characteristic 3. We hope that this collection will

interest number theorists, algebraic geometers and cryptographers and that its con-

tents will contribute to the ongoing interaction among these groups.

ix

Kristin Lauter

Ken Ribet