arXiv subjects

Noam Pirani

Publications and source records attributed to Noam Pirani.

5 recordsLinked to original sources

arXiv2026

Moments of traces of random symplectic matrices and hyperelliptic $L$-functions

We study matrix integrals of the form $$\int_{\mathrm{USp(2n)}}\prod_{j=1}^k\mathrm{tr}(U^j)^{a_j}\mathrm d U,$$ where $a_1,\ldots,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact formula (the number of terms depends only on $\sum a_j$ and not on $n,k$) for the above integral in the non-Gaussian range $\sum_{j=1}^kja_j\le 4n+1$. This extends results of Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral valid in the (Gaussian) range $\sum_{j=1}^kja_j\le n$ and $\sum_{j=1}^kja_j\le 2n+1$ respectively. We derive our formula using the connection between random symplectic matrices and hyperelliptic $L$-functions over finite fields, given by an equidistribution result of Katz and Sarnak, and an evaluation of a certain multiple character sum over the function field $\mathbb F_q(x)$. We apply our formula to study the linear statistics of eigenvalues of random unitary symplectic matrices in a narrow bandwidth sampling regime.

math.PR↗

arXiv2026

Closed geodesics in homology classes modulo sublattices

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $Γ\subset \mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $Γ$ in the large genus limit. Averaging over all lattices of index $q$, with $q \to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $Γ$, tends to $λ>0$ or $\infty$). We also study the unnormalized variance $G_M(X,Γ)$ of the counts among homology classes, and show that as $X \to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

math.NT↗

arXiv2024

Traces of powers of random matrices over local fields

Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n$, the unitary symplectic group $USp_{2n}$ or the orthogonal group $O_n$. Diaconis and Shashahani proved that the traces $\mathrm{tr}(M),\mathrm{tr}(M^2),\ldots,\mathrm{tr}(M^k)$ converge in distribution to independent normal random variables as $k$ is fixed and $n\to\infty$. Recently, Gorodetsky and Rodgers proved analogs for these results for matrices chosen from certain finite matrix groups. For example, let $M$ be chosen uniformly at random from $U_n(\mathbb{F}_q)$. They show that $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^{k}$ converge in distribution to independent uniform random variables in $\mathbb{F}_{q^2}$ as $k$ is fixed and $n\to\infty$. We prove analogs for these results over local fields. Let $\mathcal{F}$ be a local field with a ring of integers $\mathcal{O}$, a uniformizer $π$, and a residue field of odd characteristic. Let $\mathcal{K}/\mathcal{F}$ be an unramified extension of degree $2$ with a ring of integers $\mathcal{R}$. Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n(\mathcal{O})$, and fix $k$. We prove that the traces of powers $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^k$ converge to independent uniform random variables on $\mathcal{R}$, as $n\to\infty$. We also consider the case where $k$ may tend to infinity with $n$. We show that for some constant $c$ (coming from the mod $π$ distribution), the total variation distance from independent uniform random variables on $\mathcal{R}$ is $o(1)$ as $n\to\infty$, as long as $k<c\cdot n$. We also consider other matrix groups over local fields and prove similar results for them. Moreover, we consider traces of powers $M^{pi}$ and traces of negative powers, and show that apart from certain necessary modular restrictions, they also equidistribute in the limit.

math.NT↗

arXiv2023

Abhyankar's Affine Arithmetic Conjecture for the Symmetric and Alternating Groups

We prove that for any prime $p>2$, $q=p^ν$ a power of $p$, $n\ge p$ and $G=S_n$ or $G=A_n$ (symmetric or alternating group) there exists a Galois extension $K/\mathbb F_q(T)$ ramified only over $\infty$ with $\mathrm{Gal}(K/\mathbb F_q(T))=G$. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.

math.NT↗

arXiv2022

Local Statistics for Zeros of Artin-Schreier L-functions

We study the local statistics of zeros of $L$-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier $L$-functions: the ordinary, polynomial (the $p$-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.

math.NT↗