On a Theorem of Jiang and Rallis

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 x³ - 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 x³ -bx -c, with b,c in F, and computing nine of them. In this work, we compute 15 of Pleso's integrals explicitly, and reduce the last to a conjecture about the number of points on a surface over a finite field, in the special case when F is the p-adic numbers (F = Q_p) and p is equivalent to 5 mod 6. The proof of this conjecture is provided in the appendix section. 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.