arXiv subjects

Zachary Feng

Publications and source records attributed to Zachary Feng.

3 recordsLinked to original sources

arXiv2026

Non-admissibility of some universal supersingular representations

Let $K/\mathbf{Q}_p$ be an unramified extension of degree $f$ with residue field $k$. Let $\sigma$ be an irreducible representation of $\mathrm{GL}_n(k)$ over $\overline{\mathbf{F}}_p$. For $n\ge 3$, we prove that the universal supersingular representation of weight $\sigma$ is non-admissible and of infinite length when $\sigma$ is sufficiently generic and satisfies certain technical conditions. This generalizes the previous results for $n=2$ and a non-trivial finite extension $K/\mathbf{Q}_p$. Our method employs a weight cycling argument together with recent progress on the Serre weight conjectures.

math.NT↗

arXiv2025

Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

Let $\pi$ be a polarized, regular algebraic, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$ where $F$ is totally real or imaginary CM, and let $(\rho_\lambda)_\lambda$ be its associated compatible system of Galois representations. Suppose that $7\nmid n$ and, if $4\mid n$, then $n = 4p$ for some prime number $p$. We prove that there is a Dirichlet density $1$ set of rational primes $\mathcal{L}$ such that whenever $\lambda\mid \ell$ for some $\ell\in \mathcal{L}$, then $\rho_\lambda$ is irreducible.

math.NT↗

arXiv2022

A lower bound on the proportion of modular elliptic curves over Galois CM fields

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $ζ_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}(ζ_n)$ with $5\nmid n$, this proportion is at least $1-\varepsilon$ with only finitely many exceptions of $n$, for any choice of $\varepsilon > 0$.

math.NT↗