Year: 2024
Type: article
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. Thereby, 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 $(2,2)$ .
Source: Compositio Mathematica
Authors: Ilya Khayutin, Raphael S. Steiner
Institution: Northwestern University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: Research in Mathematics
OpenAlex ID: W3086819596