L-functions of Kloosterman sheaves

2021-12-27 16:00-17:00 Zoom

Abstract: Kloosterman sums are finite field analogues of Bessel functions. They appear as traces of Frobenius on some \ell-adic local systems Kl_{n+1} on G_m, called Kloosterman sheaves. In a recent work, Fresán-Sabbah-Yu have constructed some motives attached to symmetric powers of Kloosterman sheaves Sym^k Kl_{n+1}, and showed that for n=1, their L-functions have meromorphic extensions to the complex plane and satisfy functional equations. For small values of k, these L-functions are given by modular forms that one can explicitly write down. In this talk, I will present some new results about the L-functions of Sym^k Kl_3. In particular, I will specify some modular forms and explain why these motives are modular in these cases.

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