*Nicholas M. Katz*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.001.0001
- Subject:
- Mathematics, Number Theory

This book 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 ...
More

This book 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.Less

This book 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.

*Nicholas M. Katz*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0027
- Subject:
- Mathematics, Number Theory

This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for k a finite ...
More

This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for k a finite field of characteristic 2, and any character Ï‡ of kË£, the Tate-twisted Kloosterman sheaf of rank seven has Ggeom = Garith = Gâ‚‚. The first task is to express its stalk at a fixed point a É› kË£ as the finite field Mellin transform of the desired object N(a; k).Less

This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for *k* a finite field of characteristic 2, and any character Ï‡ of *k*Ë£, the Tate-twisted Kloosterman sheaf of rank seven has *G*_{geom} = *G*_{arith} = *G*â‚‚. The first task is to express its stalk at a fixed point a É› *k*Ë£ as the finite field Mellin transform of the desired object *N*(*a*; *k*).