Jurg Kramer | Searcharxiv

arXiv2013

Uniform sup-norm bounds on average for cusp forms of higher weights

Let $Γ\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d}\}$ be an orthonormal basis of $\mathcal{S}_{k}^Γ$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{k}^Γ(z):=\sum_{j=1}^{d}| f_{j}(z)|^{2}\,\mathrm{Im}(z)^{2k}$ is bounded as $O_Γ(k)$ in the cocompact setting, and as $O_Γ(k^{3/2})$ in the cofinite case, where the implied constants depend solely on $Γ$. We also show that the implied constants are uniform if $Γ$ is replaced by a subgroup of finite index.

math.NT↗

arXiv2010

An effective bound for the Huber constant for cofinite Fuchsian groups

Let $Γ$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=Γ\setminus \HH $ be the corresponding quotient space. For $γ,$ a closed geodesic of $M$, let $l(γ)$ denote its length. The prime geodesic counting function $π_{M}(u)$ is defined as the number of $Γ$-inconjugate, primitive, closed geodesics $γ$ such that $e^{l(γ)} \leq u.$ The \emph{prime geodesic theorem} implies: $$π_{M}(u)=\sum_{0 \leq λ_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) + O_{M}(\frac{u^{3/4}}{\log{u}}), $$ where $0=λ_{M,0} < λ_{M,1} <...$ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - λ_{M,j}}. $ Let $C_{M}$ be smallest implied constant so that $$|π_{M}(u)-\sum_{0 \leq λ_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}})|\leq C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u > 1.$}$$ We call the (absolute) constant $C_{M}$ the Huber constant. The objective of this paper is to give an effectively computable upper bound of $C_{M}$ for an arbitrary cofinite Fuchsian group. As a corollary we estimate the Huber constant for $\PSL(2,\ZZ),$ we obtain $C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.

math.NT↗