Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

Abstract

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.