Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms
This book is a specialized work on sums over finite fields. This algebraic topic was begun by Gauss when he considered sums over prime fields of the product of non-trivial additive and multiplicative characters of the field. From that starting point, the situations where similar sums appear are ubiquitous in Number Theory. For example, they can be found in studies devoted to the four square problem, or bounds for the number of solutions of some polynomials on finite fields.
A remarkable situation behind these objects is the distribution of the value of these sums when the size of the field goes to infinity. It turns out that they are uniformly distributed with respect to a convenient measure, a result which required a substantial investigation and that is connected with other algebraic contexts. The author is a world experto on this topic, with remarkable contributions in the literature. In fact the book under review is not the first contribution of Prof. Katz in Annals of Mathematics Studies devoted to equidistribution of sums or related problems. The interested reader can find a list of them in the webpage of Princeton University Press. In this book, the author tackles the questions in broad generality, giving a general statement on equidistribution in the modern language of sheaves.
The organization of the book is as follows. The first chapter motivates the problem and gives the statement of the main results of the book. Chapters 2 to 7 provided the constructions needed for these main results, which are enriched with complementary results in chapters 8 to 12. The rest of the chapters present a interesting variety of examples illustrating the theory whereas the final part of it addresses the situation in the case of the ring of integer numbers.
The final result is a complete book, with beautiful and important results on equidistribution properties of sums built from subtle and complicated techniques borrowed from Algebra and Geometry. This book is mainly aimed at experts on the field as well as advanced readers interested in algebraic number theory.