On a Theorem of Jiang and Rallis

TL;DR

This paper completes the explicit evaluation of a family of local integrals associated with cubic polynomials over p-adic fields, extending previous work by removing restrictions on the field and computing most integrals explicitly.

Contribution

It computes 15 out of 16 integrals for cubic polynomials over  p-adic fields without restrictions, and reduces the last to a finite field point count, advancing the explicit evaluation of these integrals.

Findings

Successfully computed 15 integrals for cubic polynomials over  p-adic fields.

Reduced the remaining integral to a point count on a surface over a finite field.

Extended previous work by removing restrictions on the base field.

Abstract

Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field $F$, when either $F$ contains three third roots of unity, or the defining polynomial is reducible. The restriction on $F$ allowed them, among other things, to reduce the case of irreducible polynomials of the form $x^3-a$. Pleso (2009) began the work of removing the restriction on $F$ by expressing the integral as a sum of $16$ integrals for the cubic polynomial $x^3 - b x - c$ with $b,c\in F$, and computing nine of them. In this work, we compute $15$ of Pleso's integrals, and reduce the last to an elementary assertion about the number of points on a surface over a finite field, in the special case when $F$ is the $p$-adic numbers, $F=\mathbb{Q}_p$, and $p$ is equivalent to $5$ mod $6$. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials, in a higher-rank setting.