arXiv subjects

Zachary Porat

Publications and source records attributed to Zachary Porat.

5 recordsLinked to original sources

arXiv2026

On the cuspidal cohomology of Iwahori congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

We investigate automorphic forms for congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ that are Iwahori at $p$. In particular, we study the cuspidal cohomology of Iwahori congruence subgroups $\mathcal{I}(3, p)$, which are comprised of matrices in $\mathrm{SL}(3, \mathbb{Z})$ that are upper triangular modulo $p$. In order to work with $\mathcal{I}(3, p)$, we generalize key results from Ash, Grayson, and Green [J.\ Number Theory 19 (1984), pp.\ 412-436]. For levels $\mathcal{I}(3, p)$ with $p \leq 227$, we found three levels with nonzero cuspidal classes and were able to compute the action of Hecke operators at two of these levels. These are the first examples of non-essentially-self-dual automorphic representations of trivial cohomological weight that are Steinberg at $p$ appearing at Iwahori level.

math.NT↗

arXiv2026

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.

math.NT↗

arXiv2025

Computations directly on the cuspidal cohomology of congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

Ash, Grayson, and Green [J. Number Theory 19 (1984), pp. 412-436] compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\mathsf{SL}(3, \mathbb{Z})$. This subspace contains the cuspidal cohomology, which is of primary interest. We extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology. Using this method, we obtain data for prime level less than 3500, finding seven additional levels at which nonzero cuspidal classes appear and calculating local factors for five of these levels.

math.NT↗

arXiv2025

Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves

Let $\mathcal{H}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p\geq 3$, and let $\mathcal{H}_g^f$ denote the $p$-rank $f$ stratum of $\mathcal{H}_g$ for $0 \leq f \leq g$. Achter and Pries note in their 2011 work that determining the number of irreducible components of $\mathcal{H}_g^f$ would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various $p$-rank strata. Our strategy involves sampling curves over finite fields and calculating their $p$-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera $g> 1$. The data also leads us to conjecture that the moduli space $\mathcal{H}^{g-2}_g$ is irreducible and suggests that $\mathcal{H}^f_g$ is irreducible for all $1 \leq f \leq g$. We conclude with a brief discussion on $\mathcal{H}^0_g$.

math.AG↗

arXiv2021

Family sizes for complete multipartite graphs

The obstruction set for graphs with knotless embeddings is not known, but a recent paper of Goldberg, Mattman, and Naimi indicates that it is quite large. Almost all known obstructions fall into four Triangle-Y families and they ask if there is an efficient way of finding or estimating the size of such graph families. Inspired by this question, we investigate the family size for complete multipartite graphs. Aside from three families that appear to grow exponentially, these families stabilize: after a certain point, increasing the number of vertices in a fixed part does not change family size.

math.CO↗