### Abstract

Convolution and Equidistributionexplores 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.

Original language | English (US) |
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Publisher | Princeton University Press |

ISBN (Print) | 9780691153308 |

State | Published - Jan 24 2012 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms*. Princeton University Press.