Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations

TL;DR

This paper characterizes when certain reducible mod ll Galois representations originate from newforms of specific levels and weights, confirming a conjecture for levels with two prime factors and providing explicit examples.

Contribution

It determines necessary and sufficient conditions for reducible mod ll Galois representations to come from newforms with two prime level factors, proving a conjecture in this setting.

Findings

Characterization of when reducible mod ll representations arise from newforms.

Proof of a conjecture by Billerey and Menares for levels with two primes.

Construction of explicit examples for levels pq with specified Atkin-Lehner eigenvalues.

Abstract

Let $k \ge 2$ be an even integer, $ \ell \ge \max\{5, k-1\} $ be a prime, and $N$ be a squarefree positive integer. It is known that if the $\rm{mod}\,\ell$ Galois representation $\overline{\rho}_f$ associated with a newform $f$ of weight $k$, level $N$, and trivial nebentypus is reducible, then $\overline{\rho}_f \simeq 1 \oplus \overline{\chi}_\ell^{k-1}$, up to semisimplification, where $\overline{\chi}_\ell^{}$ is the $\rm{mod}\,\ell$ cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the $\rm{mod}\,\ell$ representation $1 \oplus \overline{\chi}_\ell^{k-1}$ arises from a newform of weight $k$, level $N$ with exactly two prime factors with specified Atkin-Lehner eigenvalues. Specifically, this proves a conjecture of Billerey and Menares when $N$ is a product of two primes under some mild assumption. As an application, we show that for any $\ell\ge 5$ and $k=2$ or $\ell+1$, there exist a large class of distinct primes $p$ and $q$ such that the $\rm{mod}\,\ell$ representation $1 \oplus \overline{\chi}_\ell^{k-1}$ arises from a newform of weight $k$ and level $pq$ with explicit Atkin-Lehner eigenvalues.