Non-optimal levels of some reducible mod $p$ modular representations
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2022
- DOI:
- 10.48550/arXiv.2206.06209
- arXiv:
- arXiv:2206.06209
- Bibcode:
- 2022arXiv220606209D
- Keywords:
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- Mathematics - Number Theory;
- 11F33;
- 11F80 (Primary)
- E-Print:
- v3: 42 Pages, significantly improved the main results, incorporated the changes suggested by the referees. Comments are welcome