A generalisation of Zhang's local Gross-Zagier formula

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.