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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Notes
The paper could have been written using only the results from this section, however, I feel there is something to be gained from the simplicity using the first splitting function in terms of potential future work.
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This work was partially supported by a Grant from the Simons Foundation #314961.
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Haessig, C.D. L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates). Math. Ann. 369, 17–47 (2017). https://doi.org/10.1007/s00208-016-1454-6
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DOI: https://doi.org/10.1007/s00208-016-1454-6