张氏局部Gross-Zagier公式的推广
A generalisation of Zhang's local Gross-Zagier formula
摘要：
本文以GL（2）的局部Gross-Zagier公式为背景，研究了一类p-adic问题。局部Gross-Zagier公式给出了Rankin L-函数的局部傅里叶系数与非常特殊的局部几何数据（局部环绕数）的恒等式。局部环绕数是几何（高度）配对的局部系数。傅里叶系数是GL（2）的两个自守表示的局部Whittaker函数的乘积。我们建立了局部环绕数空间与所有Whittaker乘积空间的匹配。进一步，我们在局部环绕数上构造了一个通用定义的算子，它反映了解析Hecke算子的行为。它的适用性表明，从它恢复一个相当于当地的Gross-Zagier公式。我们的方法始终是建设性的和计算性的。
英文摘要：
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.
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