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L-functions of symmetric powers of cubic exponential sums

C. Douglas Haessig
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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 x^3 + 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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C. Douglas Haessig. 2008-01-09. L-functions of symmetric powers of cubic exponential sums. https://arxiv.org/abs/math/0608521

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