Non-minimal modularity lifting in weight one.

We prove an integral R = 𝐓 {R=\mathbf{T}} theorem for odd two-dimensional <italic>p</italic>-adic representations of G 𝐐 {G_{\mathbf{Q}}} which are unramified at <italic>p</italic>, extending results of [<xref>5</xref>] to the non-minimal case. We prove, for any <italic>p</italic>, the existence of Katz modular forms modulo <italic>p</italic> of weight one which do not lift to characteristic zero. [ABSTRACT FROM AUTHOR]