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, ${\overline{\rho }}_0$ be a reducible, odd and semi-simple representation of $G_{Q,Np}$ 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 ${\overline{\rho }}_0$ such that f is new at every ${\ell }_i$. Moreover, suppose $p|{\ell }_{i_0}+1$ 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 ${\overline{\rho }}_0$. As a consequence, we prove a conjecture of Billerey–Menares in many cases.