L-functions of symmetric powers of cubic exponential sums

2006/08/21 by C. Douglas Haessig, Haessig, C. Douglas
Mathematics · #11L03 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11L03

paper · pdf · doi:10.48550/arxiv.math/0608521

Implemented changes suggested by referee. Fixed typos and improved exposition of some sections. Proof of Conjecture 3.1 given. 38 pages

arxiv created 2008/01/09 · arxiv updated 2009/12/01

Abstract

For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F-crystal associated to the family of cubic exponential sums of x³ + λx. We explore its rationality, field of definition, degree, trivial factors, functional equation, and Newton polygon. The paper is essentially self-contained, due to the remarkable and attractive nature of Dwork's p-adic theory. A novel feature of this paper is an extension of Dwork's effective decomposition theory when k < p. This allows for explicit computations in the associated p-adic cohomology. In particular, the action of Frobenius on the (primitive) cohomology spaces may be explicitly studied.

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