Period functions for vector-valued Maass cusp forms of real weight, with an application to Jacobi Maass cusp forms

2024/08/06 by Pohl, Anke, Choie, YoungJu, Bruggeman, Roelof
#11F50 11F67 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.03104

Abstract

For vector-valued Maass cusp forms for~SL₂(ℤ) with real weight~k∈ℝ and spectral parameter s∈ℂ, Re s∈ (0,1), s\not≡ ± k/2 mod 1, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue 1 of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight k+1/2 for the semi-direct product of SL₂(ℤ) with the integer points Hei(ℤ) of the Heisenberg group.

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