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

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 $(2,2)$.

Clarifications and corrections

References in corpus (2)

• Theta functions, fourth moments of eigenforms, and the sup-norm problem II
• Small diameters and generators for arithmetic lattices in ${\mathrm{SL}}_2(\mathbb{R})$ and certain Ramanujan graphs

Cited by in corpus (3)

• Theta functions, fourth moments of eigenforms, and the sup-norm problem II
• Beyond the spherical sup-norm problem
• $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity