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

Abstract

Let $p\ge 5$ 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 $1\le i_0\le r$. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level $N{\ell }_{i_0}^2{\prod }_{j\ne i_0}{\ell }_j$ 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.

arXiv commentv3: 42 Pages, significantly improved the main results, incorporated the changes suggested by the referees. Comments are welcome