Vol. 315, No. 1, 2021

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The index of a modular differential operator

William Duke

Vol. 315 (2021), No. 1, 45–73

The space of all weakly holomorphic modular forms and the space of all holomorphic period functions of a fixed weight for the modular group are realized as locally convex topological vector spaces that are topologically dual to each other. This framework is used to study the kernel and range of a linear differential operator that preserves modularity and to define and describe its adjoint. The main results are an index formula for such a differential operator that is holomorphic at infinity and the identification of the cokernel of the operator as a cohomology group of the modular group acting on the kernel.

differential operator
Mathematical Subject Classification
Primary: 11F11
Secondary: 34M03
Received: 6 October 2020
Revised: 12 February 2021
Accepted: 25 September 2021
Published: 13 December 2021
William Duke
Mathematics Department
Los Angeles, CA
United States