TY - JOUR
AU1 - Deo, Shaunak V.
AB - 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.
TI - Non-optimal levels of some reducible mod $p$ modular representations
JF - Mathematics
DO - 10.48550/arxiv.2206.06209
DA - 2022-06-13
UR - https://www.deepdyve.com/lp/arxiv-cornell-university/non-optimal-levels-of-some-reducible-mod-p-modular-representations-TTvtKp2Ccm
VL - 2024
IS - 2206
DP - DeepDyve
ER -