arXiv subjects

Petru Constantinescu

Publications and source records attributed to Petru Constantinescu.

4 recordsLinked to original sources

arXiv2025

Non-vanishing of geodesic periods of automorphic forms

We prove that one hundred percent of the closed geodesic periods of a Hecke--Maaß cusp form for the modular group are non-vanishing when ordered by length. We present applications to the non-vanishing of central values of Rankin--Selberg $L$-functions. Similar results for holomorphic forms for general Fuchsian groups of finite covolume with a cusp are also obtained, as well as results towards normal distribution. Our new key ingredient is to relate the distributions of closed geodesic periods and vertical line integrals via graph theory.

math.NT↗

arXiv2021

Dissipation of correlations of holomorphic cusp forms

We obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight. Our new ingredient is to incorporate the spectral theory of weight $k$ automorphic functions to the method of Holowinsky-Soundararajan. For Holowinsky's shifted convolution sums approach, we need to develop new bounds for the Fourier coefficients of weight $k$ cusp forms. For Soundararajan's subconvexity approach, we use Ichino's formula for evaluating triple product integrals.

math.NT↗

arXiv2021

Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic $n$-space

We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.

math.NT↗

arXiv2020

Distribution of modular symbols in $\mathbb{H}^3$

We introduce a new technique for the study of the distribution of modular symbols, which we apply to congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of $\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups, where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and the second moment of the distribution.

math.NT↗