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.