## Convolution and Equidistribution: |

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. "The book is written in a clear and enlightening style. The author provides the reader with many examples that are developed throughout a dozen chapters. These examples help understand and clarify the depth and the variety of applications of the beautiful main equidistribution statement that relies on rather complicated and subtle algebrageometric arguments." "The book provides the reader with much material around the question of the equidistribution of the angles if one fixes "Once a certain basic understanding is reached, this book, like the others written by N. Katz, reveals itself to be very precisely and sharply written, and to be full of riches. And finally, this theory shows spectacularly how some of the most abstract ideas of algebra and algebraic geometry may be essential to solving extremely concrete problems."
- Arithmetic Moduli of Elliptic Curves. (AM-108). [Paperback]
- Exponential Sums and Differential Equations. (AM-124). [Paperback]
- Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116). [Paperback]
- Moments, Monodromy, and Perversity. (AM-159): A Diophantine Perspective. (AM-159). [Paperback]
- Rigid Local Systems. (AM-139). [Paperback]
- Twisted L-Functions and Monodromy. (AM-150). [Paperback]
- Annals of Mathematics Studies
Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Editors
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