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

This article examines the Fourier expansions of certain non-classical <em>p</em>-adic modular forms of weight one: the eponymous <em>generalised eigenforms</em> 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 <em>K</em> in which the prime <em>p</em> splits, the coe!cients of the attendant generalised eigenform are expressed as <em>p</em>-adic logarithms of algebraic numbers belonging to an idoneous ring class field of <em>K</em>. 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 <em>p</em>-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.