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
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