论文标题
某些还原的mod $ p $模块化表示的非最佳级别
Non-optimal levels of some reducible mod $p$ modular representations
论文作者
论文摘要
令$ p \ geq 5 $为素数，$ n $是$ p $，$ \barρ_0$不可分解的整数，是$ g _ {\ mathbb {q}的$ g _ {\ mathbb {q}，np { $ np $。在假设某个Selmer组的尺寸最多为$ 1 $之后，我们发现存在足够的条件，可以使cuspidal eigenform $ f $ a级$ n \ prod_ {i = 1}^{r}^{r} \ ell_i $和适当的flowerting liftting $ \barρ_0$ \barρ_0$，这样的$ f $是每个$ f $ at ever $ \ ell_e el_e el_ el_e el_i $ \ ell_i $。此外，假设$ p \ mid \ ell_ {i_0}+1 $对于某些$ 1 \ leq i_0 \ leq r $。然后，在假设某个Selmer群体消失后，我们发现存在足够的条件，可以存在$ n \ ell_ \ ell_ {i_0}^2 \ prod_ {j \ neq I_0} \ ell_j $和适当的重量的cuspidal特征形式，在每一个$ \ ell_i $ and tups $ \barρ_0都是新的。结果，我们证明了Billerey的猜想 - 在许多情况下。
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.
