Consequences of the Gross/Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p

TL;DR

This paper explores the implications of Gross/Zagier formulae on the stability, subconvexity, and non-vanishing of average L-values of holomorphic forms, connecting these to class numbers of imaginary quadratic fields.

Contribution

It provides an exact average formula for twisted L-values, demonstrating stability, subconvexity bounds, and non-vanishing results in the classical setting.

Findings

Derived an exact average formula for L-values

Established stability of average L-values

Proved subconvexity bounds and non-vanishing results

Abstract

In this paper we investigate some consequences of the Gross/Zagier types of formulae which were introduced by Gross and Zagier, and were then generalized in various directions by Hatcher, Zhang, Kudla and several other people. Working in the classical context of central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. One also obtains a stability result, as well as subconvexity (in this setting), and certian non-vanishing assertions. This article is dedicated to the memory of Serge Lang.