Supersingular Abelian Varieties Over Finite Fields

Abelian varieties can be classified via their moduli. In positive characteristic the structure of the p-torsion-structure is an additional, useful tool. For that structure supersingular abelian varieties can be considered the most special ones. They provide a starting point for the fine description of various structures. For low dimensions the moduli of supersingular abelian varieties is by now well understood. In this book we provide a description of the supersingular locus in all dimensions, in particular we compute the dimension of it: it turns out to be equal to Äg.g/4Ü, and we express the number of components as a class number, thus completing a long historical line where special cases were studied and general results were conjectured (Deuring, Hasse, Igusa, Oda-Oort, Katsura-Oort).

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.

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.

Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. The papers included in this publication introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension of at least 2.

This book presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory, and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.