# Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations?

Let $$p$$ be a prime and $$\mathbb{F}$$ a finite field of characteristic $$p$$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $$\bar{\rho}:G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F})$$ which is ramified at finitely many primes lifts to a modular Galois representation associated to an eigenform of optimal weight and level. The optimal level of this form is the prime to $$p$$ part of the Artin conductor of $$\bar{\rho}$$. This was the strong form of Serre's conjecture.

Hamblen and Ramakrishna showed that the hypothesis of irreducibility may be relaxed by showing that under some conditions, a reducible and indecomposable $$\bar{\rho}$$ lifts to a Galois representation associated to an eigenform. However, they are not able to optimize the level of the eigenform.

My question is the following: is it expected that the optimal level is the prime to $$p$$ part of the Artin conductor of $$\bar{\rho}$$? It may be the case that it is not exactly this, and if it is not, is there an explicit counterexample to this?

• It would be of interest I feel to have some data for levels when $\bar{\rho}$ is indecomposable and not unipotent, this is roughly the case in which the weak form of Serre's conjecture is known. – user130124 Nov 4 '18 at 6:21

Billerey and Menares have studied this question for the reducible representations $$\bar{\rho} = 1 \oplus \chi_p^{k-1}$$ in https://arxiv.org/abs/1309.3717 In this case the prime-to-$$p$$ part of the Artin conductor is 1, and it is not always the case that $$\bar{\rho}$$ arises from a cuspidal eigenform of weight $$k$$ and level $$1$$.
• For $k=2$, I would also suggest to look at the work of Hwajong Yoo (and Ribet). See arxiv.org/pdf/1409.8342.pdf – Emmanuel Lecouturier Oct 21 '18 at 2:09
In addition to François' answer, here is what can be said if you only seek for an isomorphism between the semi-simplification of your residual representation $$\overline{\rho}$$ and the semi-simplification of the reduction of a $$p$$-adic representation attached a Hecke eigenform.
Consider an odd mod $$p$$ Galois representation $$\overline{\rho}=\nu_1\oplus\nu_2$$ of Serre weight $$k$$ and level $$N$$ (coprime to $$p$$) and assume $$p>k+1$$. Then, there exist $$\epsilon_1,\epsilon_2$$ two Galois characters unramified at $$p$$ such that $$\overline{\rho}\simeq\epsilon_1\oplus\epsilon_2\chi_p^{k-1}$$. Set $$\eta=\epsilon_1^{-1}\epsilon_2$$. Then, there is a newform $$f$$ of (optimal) weight $$k$$ and level $$N$$ and a prime ideal $$\mathfrak{p}$$ over $$p$$ in $$\overline{\mathbf{Q}}$$ such that we have $$\overline{\rho}\simeq\overline{\rho}_{f,\mathfrak{p}}^{ss}$$ if and only if we have $$B_{k,\eta}=0$$ or $$\eta(\ell)\ell^k=1$$ for some prime $$\ell$$ dividing $$N$$.
Here, $$\eta(\ell)=\eta(\mathrm{Frob}_\ell)$$ if $$\eta$$ is unramified at $$\ell$$ and $$\eta(\ell)=0$$ otherwise. (Roughly speaking $$B_{k,\eta}$$ is the mod $$p$$ reduction of the $$k$$-th Bernoulli number associated with a lift of $$\eta$$.)