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

2009/08/09 by C. Douglas Haessig, Haessig, C. Douglas, Antonio Rojas-Leon +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.0908.1240

This paper has been withdrawn by the authors.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

arxiv created 2010/08/02 · arxiv updated 2010/08/04

Abstract

For ψa nontrivial additive character on the finite field F_q, the map t ↦ ∑_{x ∈ Fq} ψ(f(x)+tx) is the Fourier transform of the map t ↦ ψ(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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