L-functions of symmetric powers of cubic exponential sums

Details L-functions of symmetric powers of cubic exponential sums L-functions of symmetric powers of cubic exponential sums

2006-08-21

arxiv.org/abs/math/0608521v4

Mathematics

Number Theory

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

Scientific paper

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 x^3 + \lambda x. 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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