Equidistribution of Kloosterman sums over function fields

TL;DR

The paper advances the horizontal Sato–Tate understanding for Kloosterman sums over function fields by proving explicit-error equidistribution results when the corresponding places vary in arithmetic progressions and short intervals, and by establishing joint Sato–Tate distributions for families of such sums. It leverages the Kloosterman sheaf with $\mathrm{SL}_2$ monodromy, symmetric-power functoriality, and effective Chebotarev bounds, together with Weyl-type Fourier expansions and an involution linking short intervals to progressions. The main contributions are explicit error terms of the form $O(q^{-d/4}\sqrt{[E:K]\mathfrak N_{a,E/K}})$ in progression settings, a short-interval analogue with $O(q^{(d-2h-2)/4}...)$ bounds in the rational function field, and a joint distribution framework yielding Erdős–Turán-type error terms for finite families of Kloosterman sums. These results deepen the quantitative understanding of Sato–Tate phenomena in function-field settings and have potential implications for the study of $L$-functions and monodromy in arithmetic geometry.

Abstract

We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved.

Theorems & Definitions (38)

• Theorem 1.1: Katz, Ka88
• Conjecture 1.1: Katz, Ka80
• Theorem 1.2: Chai--Li, CL03
• Theorem 1.3
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.4
• Theorem 1.5