Non-optimal levels of some reducible mod $p$ modular representations

TL;DR

This paper establishes conditions for the existence of certain cuspidal eigenforms with prescribed levels and weights that lift reducible mod p representations, proving a conjecture of Billerey--Menares in many cases.

Contribution

It provides new sufficient conditions for the existence of cuspidal eigenforms lifting reducible mod p representations with specified levels, addressing a conjecture by Billerey--Menares.

Findings

Conditions for eigenforms with level divisible by primes not dividing N.

Existence of eigenforms with level squared at a prime dividing the level.

Proof of Billerey--Menares conjecture in many cases.

Abstract

Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\bar\rho_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of primes not dividing $Np$. After assuming that a certain Selmer group has dimension at most $1$, we find sufficient conditions for the existence of a cuspidal eigenform $f$ of level $N\prod_{i=1}^{r}\ell_i$ and appropriate weight lifting $\bar\rho_0$ such that $f$ is new at every $\ell_i$. Moreover, suppose $p \mid \ell_{i_0}+1$ for some $1 \leq i_0 \leq r$. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level $N\ell_{i_0}^2 \prod_{j \neq i_0} \ell_j$ and appropriate weight which is new at every $\ell_i$ and which lifts $\bar\rho_0$. As a consequence, we prove a conjecture of Billerey--Menares in many cases.