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

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 $\mathrm{mod}\mathrm{\ell }$ 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 $\mathrm{mod}\mathrm{\ell }$ cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the $\mathrm{mod}\mathrm{\ell }$ 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 $\mathrm{mod}\mathrm{\ell }$ representation $1\oplus {\overline{\chi }}_{\ell }^{k-1}$ arises from a newform of weight $k$ and level $pq$ with explicit Atkin-Lehner eigenvalues.

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