Convolution and Equidistribution explores an important aspect of number theory–the theory of exponential sums over finite fields and their Mellin transforms–from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.
The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book’s main theorem, using a mixture of geometric, categorical, and group-theoretic methods.
By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.
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Nicholas M. Katz is professor of mathematics at Princeton University. He is the author or coauthor of six previous titles in the Annals of Mathematics Studies:Arithmetic Moduli of Elliptic Curves (with Barry Mazur);
Gauss Sums, Kloosterman Sums, and Monodromy Groups;
Exponential Sums and Differential Equations;
Rigid Local Systems;
Twisted L-Functions and Monodromy; and
Moments, Monodromy, and Perversity.
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