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

by Ilya Khayutin, Paul D. Nelson, and Raphael S. Steiner.

Two Hecke eigenforms on $S^2$.

Abstract

Let $f$ be an $L^2$-normalized holomorphic newform of weight $k$ on $\Gamma_0(N) \backslash \mathbb{H}$ with $N$ squarefree, or more generally, on any hyperbolic surface $\Gamma \backslash \mathbb{H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$. Denote by $V$ the hyperbolic volume of said surface. We prove the sup-norm estimate

with absolute implied constant. For a cuspidal Maaß newform $\varphi$ of eigenvalue $\lambda$ on such a surface, we prove that

We establish analogous estimates in the setting of definite quaternion algebras.

Links

arXiv

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Twisted Linnik implies optimal covering exponent for $S^3$

by Tim Browning, V. Vinay Kumaraswamy, and Raphael S. Steiner.

Points on the **two-dimensional** sphere obtained by five translations of the north pole by the Golden Gates of Sarnak, $ \left\{ \frac{1}{\sqrt{5}} \left( \begin{smallmatrix} 1+2i & 0 \\ 0 & 1-2i \end{smallmatrix} \right), \frac{1}{\sqrt{5}} \left( \begin{smallmatrix} 1 & 2i \\ 2i & 1 \end{smallmatrix} \right), \frac{1}{\sqrt{5}} \left( \begin{smallmatrix} 1 & 2 \\ -2 & 1 \end{smallmatrix} \right) \right\}$$\subset \mathrm{SU}_2 \curvearrowright S^2$, or their inverses.

Abstract

We show that a twisted variant of Linnik’s conjecture on sums of Kloosterman sums leads to an optimal covering exponent for $S^3$.

Links

International Mathematics Research Notices, Vol. 2019 1,
arXiv

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Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs

by Raphael S. Steiner.

The hyperbolic surface corresponding to a maximal order in a quaternion algebra over $\mathbb{Q}$ of reduced discriminant 35 depicted in the Poincaré disk. The image was computed by James Rickards.

Abstract

We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a “small” covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.

Links

The Ramanujan Journal
arXiv

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