Congruences of local origin and automorphic induction

Abstract

We explore the possibilities for the Galois representation [Formula: see text] attached to a weight-one newform [Formula: see text] to be residually reducible, i.e. for the Hecke eigenvalues to be congruent to those of a weight-one Eisenstein series. A special role is played by Eisenstein series [Formula: see text] of level [Formula: see text], where [Formula: see text] is the quadratic character associated with an imaginary quadratic field [Formula: see text], of discriminant [Formula: see text], with respect to which [Formula: see text] is of dihedral type. We prove congruences, where the modulus divides either the class number [Formula: see text] or [Formula: see text] (for a prime [Formula: see text]), and [Formula: see text] is of level [Formula: see text] in the first case, level [Formula: see text] or [Formula: see text] (according as [Formula: see text] or [Formula: see text] respectively) in the second. We also prove analogous congruences where [Formula: see text] and [Formula: see text] are replaced by a newform [Formula: see text] and its twist by [Formula: see text], and [Formula: see text] is replaced by a Siegel cusp form of genus [Formula: see text] and paramodular level, induced in some sense from a Hilbert modular form.