![]() | |
| "This is a monograph in the Annals of Mathematics series that studies the monodromy groups, associated to certain local systems arising from hypergeometric sheaves (and with some relationship with exponential sums). In the book, the authors first deduce various conditions on the monodromy group associated to the local system to get some quite powerful classification theorems. Some features of the book include: (a) introductory material on monodromy groups; (b) various results translating the information on hypergeometric sheaves and local system to group theory; (c) a summary of some results in representation theory which are not available in book form; (d) new results on irreducible subgroups of classical groups containing special elements, which will have applications in many different areas; and (e) results on moments and deciding when the monodromy group is infinite and estimating its size when it is finite"-- "An examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fieldsExponential 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"-- |
|
|
