A generalisation of Zhang's local Gross-Zagier formula

TL;DR

This paper extends Zhang's local Gross-Zagier formulae for GL(2) by exploring p-adic problems, establishing a correspondence between local geometric data and Fourier coefficients, and constructing an operator reflecting Hecke operator behavior.

Contribution

It introduces a universal operator on local linking numbers and demonstrates its effectiveness in recovering local Gross-Zagier formulae, with a constructive and computational approach.

Findings

Established a matching between local linking numbers and Whittaker product spaces.

Constructed a universal operator reflecting Hecke operator behavior.

Recovered local Gross-Zagier formulae using the new operator.

Abstract

On the background of Zhang's local Gross-Zagier formulae for GL(2), we study some p-adic problems. The local Gross-Zagier formulae give identities of very special local geometric data (local linking numbers) with certain local Fourier coefficients of a Rankin L-function. The local linking numbers are local coefficients of a geometric (height) pairing. The Fourier coefficients are products of the local Whittaker functions of two automorphic representations of GL(2). We establish a matching of the space of local linking numbers with the space of all those Whittaker products. Further, we construct a universally defined operator on the local linking numbers which reflects the behavior of the analytic Hecke operator. Its suitability is shown by recovering from it an equivalent of the local Gross-Zagier formulae. Our methods are throughout constructive and computational.