L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates)

2015/04/22 by C. Douglas Haessig, Haessig, C. Douglas · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1504.05802

18 pages. Exposition improved. Many typos corrected

openalex publication_date 2015/04/22 · arxiv created 2016/05/18 · arxiv updated 2016/05/19 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these roots. Our method is indirect. We first develop a Dwork-type p-adic cohomology theory for the two-variable infinite symmetric power L-function associated to the Kloosterman family, and then study p-adic estimates of the eigenvalues of Frobenius. A continuity argument then provides the desired p-adic estimates.

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• Infinite symmetric power L-functions of the hyper-Kloosterman family

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