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arXiv · 2009.07194

Theta functions, fourth moments of eigenforms, and the sup-norm problem I

Ilya Khayutin·Raphael S. Steiner
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Abstract

We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf{G} \times \mathbf{G} \times \mathbf{SL}_{2}$, for $\mathbf{G}$ an indefinite inner-form of $\mathbf{SL}_2$ over $\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.

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Ilya Khayutin, Raphael S. Steiner. 2020-09-15. Theta functions, fourth moments of eigenforms, and the sup-norm problem I. https://doi.org/10.1112/s0010437x24007437

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