  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.
