Non-optimal levels of a reducible mod ℓ modular representation

Yoo, Hwajong

doi:10.1090/tran/7314

Non-optimal levels of a reducible mod $\ell$ modular representation
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Trans. Amer. Math. Soc. 371 (2019), 3805-3830 Request permission

Abstract:

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $\Gamma _0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.

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