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Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups by Nicholas M. Katz at Abbey's Bookshop,
Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
— —
Nicholas M. Katz Pham Huu Tiep

Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups

Nicholas M. Katz,  Pham Huu Tiep

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English
Princeton University Press
01 October 2025
Groups & group theory; Number theory
Series: Annals of Mathematics Studies
An examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fields

Exponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck's creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne's theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions.

This book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.
By:   ,
Imprint:   Princeton University Press
Country of Publication:   United States
Dimensions:   Height: 235mm,  Width: 156mm, 
ISBN:   9780691272252
ISBN 10:   0691272255
Series:   Annals of Mathematics Studies
Pages:   594
Publication Date:  
Audience:   College/higher education ,  Professional and scholarly ,  Primary ,  Undergraduate
Format:   Paperback
Publisher's Status:   Active

Nicholas M. Katz is professor of mathematics at Princeton University. His books include Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (Princeton). Pham Huu Tiep is the Joshua Barlaz Professor and Distinguished Professor of Mathematics at Rutgers University.

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