Selberg Integral over Local Fields

TL;DR

This paper formulates and proves a Selberg integral formula for local fields of characteristic zero, extending classical and complex analogs to a broader mathematical setting.

Contribution

It introduces a novel Selberg integral formula for local fields of characteristic zero, filling a gap in the existing analogs over various fields.

Findings

Established the Selberg integral formula for local fields of characteristic zero

Extended the scope of Selberg integrals beyond classical and complex cases

Provided a new tool for analysis over local fields

Abstract

Selberg introduced his beautiful integral formula in 1944, see [Sel]. Evans [E1] conjectured a finite field analog of Selberg integral formula in 1980. And Anderson [An] proved a major case of it in 1981 and his ideas was used to obtained the complete result [E2]. On the other hand, Aomoto [Ao] proved an analog of Selberg integral for complex field ${\mathbb{C}}$ in 1987. The purpose of the present paper is to formulate and prove Selberg integral formula for local fields of characteristic zero.