Related papers: Low-lying zeros for L-functions associated to Hilb…
Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,\pi)$…
In this paper, we prove some one level density results for low-lying zeros of families of $L$-functions. More specifically, the families under consideration are that of $L$-functions of holomorphic Hecke eigenforms of level 1 and weight $k$…
We study a new orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. To illustrate our methods, we prove a one level density result for this family with the support of the…
We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and…
One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec [FI] proved that the 1-level density for L-functions…
We study one-level and two-level densities for low lying zeros of symmetric power L-functions in the level aspect. It allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign…
In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances…
It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the…
This paper is devoted to a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. Assuming the Riemann Hypothesis and the ratio conjecture,…
In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke $L$-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary,…
We calculate the one-level density of thin subfamilies of a family of Hecke cuspforms formed by twisting the forms in a smaller family by a character. The result gives support up to 1, conditional on GRH, and we also find several of the…
Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of…
The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups…
In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the…
We compute the one-level density for the family of cubic Dirichlet $L$-functions when the support of the Fourier transform of a test function is in $(-1,1)$. We also establish the Ratios conjecture prediction for the one-level density for…
We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that…
We study the distribution of values of automorphic $L$-functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula…
In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term,…
Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary…
We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real…