The author proves an integral \(R = \mathbb{T}\) theorem for weight one modular forms of non-minimal level, under certain technical assumptions (including that the residual mod \(p\) representation is unramified at \(p\)). For weight one modular forms of minimal level this was done in [the author and D. Geraghty, Invent. Math. 211, No. 1, 297–433 (2018; Zbl 1476.11078)], so the innovation of this paper is to extend this to non-minimal level. This reproves many cases of a modularity theorem due to [K. Buzzard and R. Taylor, Ann. Math. (2) 149, No. 3, 905–919 (1999; Zbl 0965.11019)].
Note that \(R\) and \(\mathbb{T}\) in this setting can be entirely \(p\)-torsion – indeed, a supplementary result of this article shows that, for every prime \(p\), there is a Katz mod \(p\) modular form of weight one and level coprime to \(p\) that does not lift to characteristic zero. This causes problems for the existing approaches to the non-minimal case, which proceed either via Wiles’s numerical criterion or a method of Khare-Wintenberger. The author ingeniously adapts the latter method. From the paper:
“The usual technique for showing that the support of \(M_\infty\) is spreadover all components is to produce modular lifts with these properties. In our context this is not possible: there are no weight one forms in characteristic zero which are Steinberg at a finite place \(q\) […]. Our replacement for producing modular points in characteristic zero is to work on the special fibre, and to show that \(M_\infty\) is (in some sense) spread out as much as possible over \(R^{1, \square}_\infty/\varpi\). [W]e do this … by working in weight \(p\) and then descending back to weight one using the doubling method.”