arXiv subjects

Peter Zenz

Publications and source records attributed to Peter Zenz.

4 recordsLinked to original sources

arXiv2026

Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle ϕ, φ^2 \rangle$ uniformly in $t_ϕ\to \infty$ and $t_φ\to \infty$, where $t_ϕ$ and $t_φ$ are the eigenparameters corresponding to Maass forms $ϕ$ and $φ$ on $X$.

math.DS↗

arXiv2025

Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity

We let $f$ be a half-integral weight modular form of weight $κ>4$ on $Γ_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = Λ_f(n)n^{\frac{κ-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of $f$. We study a weighted second moment of the central value of the $L$-function associated to $f$ over an orthogonal basis $H_κ(4)$ of $S_κ(Γ_0(4))$. This corresponds to studying the following sum: $$\sum_{f\in H_κ(4)}\frac{Λ_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$ Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound $$L(1/2,f)\ll_{\varepsilon} (κ^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$ This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of $L$-functions.

math.NT↗

arXiv2021

Quantum variance for holomorphic Hecke cusp forms on the vertical geodesic

We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth, compactly supported test functions. The variance is related to an averaged shifted-convolution problem that we evaluate asymptotically. We encounter an off-diagonal term that matches exactly with a certain diagonal term, a feature reminiscent of moments of $L$-functions.

math.NT↗

arXiv2021

Sharp bound for the fourth moment of holomorphic Hecke cusp forms

We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.

math.NT↗