C. Douglas Haessig

L-functions of symmetric powers of the generalized Airy family of exponential sums

For \psi a nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} \psi(f(x)+tx) is the Fourier transform of the map t \mapsto \psi(f(t))

L-functions of symmetric powers of cubic exponential sums

. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study the p-adic absolute values of the roots.

Meromorphy of the rank one unit root L-function revisited

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 where λ runs over . 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 Dworks p-adic theory. A novel feature of this paper is an extension of Dworks 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.

On the p-adic Riemann hypothesis for the zeta function of divisors

We demonstrate that Wans alternate description of Dworks unit root L-function in the rank one case may be modified to give a proof of p-adic meromorphy that is classical, eliminating the need to study sequences of uniform meromorphic functions.