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

In the prequel, a sharp bound in the level aspect on the fourth moment of Hecke--Maa{\ss} forms with an inexplicit (in fact exponential) dependency on the eigenvalue was given. In this paper, we develop further the framework of explicit theta test functions in order to capture the eigenvalue more precisely. We use this to reduce a sharp hybrid fourth moment bound to an intricate counting problem. Unconditionally, we give a hybrid bound, which is sharp in the level aspect and with a slightly larger than convex dependency on the eigenvalue.Comment: 28 page

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

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 $(2,2)$.Comment: Updated following comment

Sums of Cusp Form Coefficients Along Quadratic Sequences

Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 2$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X + O_{f,h,\epsilon}(X^{\frac{6k-8}{8k-11}+\epsilon})$ for $\epsilon>0$, which improves a result of Blomer from 2008 with error $X^{6/7+\epsilon}$. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.Comment: 22 pages, with a 14 page appendix from Raphael S. Steiner. This version corrects a mistake in the previous, where lifts of holomorphic modular forms to Maass forms were omitte