Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

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.

Book Details

• Author: Nicholas M. Katz
• Publisher: Princeton University Press
• eBook ISBN: 9781400842704, 1400842700
• Print ISBN: 9780691153308, 0691153302
• Other ISBNs: 9780691153315