The exotic inverted Kloosterman sum

2024/07/04 by Fu, Lei, Wan, Daqing
#11L05 #14F20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.03599

Abstract

Let B be a product of finitely many finite fields containing \mathbb F_q, ψ:\mathbb F_q→ \mathbb Q_ℓ^* a nontrivial additive character, and χ: B^*→ \mathbb Q_ℓ^* a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum EIK(\mathbb F_q, a):=∑_\substackx∈ B^*
Tr_{B/\mathbb Fq}(x)\not =0
N_{B/\mathbb Fq}(x)=a χ(x)ψ(\frac1Tr_{B/\mathbb Fq}(x)), a∈ \mathbb F_q^*. We estimate this sum using ℓ-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most 2(n+1) and mixed of weight at most n, where n+1 = dim_{\mathbb Fq}B. Up to a trivial main term, this gives the expected square root cancellation.

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