L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods

Details L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods

2009-08-09

arxiv.org/abs/0908.1240v2

Mathematics

Number Theory

This paper has been withdrawn by the authors. This paper has been updated and split into two papers

Scientific paper

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))$. 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.

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