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In this article, we describe a geometric method to study cusp forms, which relies on heat kernel and Bergman kernel analysis. This new approach of applying techniques coming from analytic geometry is based on the micro-local analysis of the heat kernel and the Bergman kernel from [3] and [2], respectively, using which we derive sup-norm bounds for cusp forms of integral weight, half-integral weight, and real weight associated to a Fuchsian subgroup of first kind.

Heat Kernels, Bergman Kernels, and Cusp Forms