Non-optimal levels of some reducible mod $p$ modular representations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913214080483328 |
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| author | Deo, Shaunak V. |
| author_facet | Deo, Shaunak V. |
| contents | Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\barρ_0$ 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 $\barρ_0$ 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 $\barρ_0$. As a consequence, we prove a conjecture of Billerey--Menares in many cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_06209 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-optimal levels of some reducible mod $p$ modular representations Deo, Shaunak V. Number Theory 11F33, 11F80 (Primary) Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\barρ_0$ 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 $\barρ_0$ 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 $\barρ_0$. As a consequence, we prove a conjecture of Billerey--Menares in many cases. |
| title | Non-optimal levels of some reducible mod $p$ modular representations |
| topic | Number Theory 11F33, 11F80 (Primary) |
| url | https://arxiv.org/abs/2206.06209 |