Computing Abelian Varieties Over Finite Fields

From Gauss to G|del, mathematicians have sought an efficient algorithm to distinguish prime numbers from composite numbers. This book presents a random polynomial time algorithm for the problem. The methods used are from arithmetic algebraic geometry, algebraic number theory and analyticnumber theory. In particular, the theory of two dimensional Abelian varieties over finite fields is developed. The book will be of interest to both researchers and graduate students in number theory and theoretical computer science.

Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.

Mots-clés de l'auteur: abelian varieties ; isogenies ; polarizations ; Mumford's theory of theta functions ; public key cryptography ; discrete logarithm problem.

The set A(k) of rational points of an abelian variety A defined over a finite field k forms a finite abelian group. This group is suitable for multiple applications, and its structure is very important. Knowing the possible group structures of A(k) and some statistics is then fundamental. In this thesis, we focus our interest in "cyclic varieties", i.e. abelian varieties defined over finite fields with cyclic group of rational points. Isogenies give us a coarser classification than that given by the isomorphism classes of abelian varieties, but they provide a powerful tool in algebraic geometry. Every isogeny class is determined by its Weil polynomial. We give a criterion to characterize "cyclic isogeny classes", i.e. isogeny classes of abelian varieties defined over finite fields containing only cyclic varieties. This criterion is based on the Weil polynomial of the isogeny class.From this, we give bounds on the fractions of cyclic isogeny classes among certain families of isogeny classes parameterized by their Weil polynomials.Also we give the proportion of "local"-cyclic isogeny classes among the isogeny classes defined over the finite field mathbb{F}_q with q elements, when q tends to infinity.

This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

This volume contains the proceedings of the 17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory (AGC2T-17), held from June 10–14, 2019, at the Centre International de Rencontres Mathématiques in Marseille, France. The conference was dedicated to the memory of Gilles Lachaud, one of the founding fathers of the AGC2T series. Since the first meeting in 1987 the biennial AGC2T meetings have brought together the leading experts on arithmetic and algebraic geometry, and the connections to coding theory, cryptography, and algorithmic complexity. This volume highlights important new developments in the field.