Level lowering for weight 1 forms

2012-10-18T17:01:01Z

I'm interested in knowing to what extent is level lowering known to hold in weight 1. Specifically, let's say I have an eigenform $f$ in $S_1(N,\chi)$ and a prime $p$ which doesn't divide $N$. Let's assume that the mod $p$ Galois representation attached to $f$ is unramified at some prime $q$ dividing $N$. Is there then an eigenform $g$ in $S_1(N/q,\chi)$ with the same mod $p$ Galois representation as $f$?

I think I can show this if $p>5$, but the argument relies on Artin's conjecture and feels like overkill. Let me sketch the argument, and then ask some more specific questions at the end.

Here's the argument: if $f$ is CM, check directly that no such $q$ can exist. So the projective image of $\overline{\rho}_f$ must be $A_4$, $S_4$, or $A_5$. In particular, if $p>5$ then one can lift $\overline{\rho}_f$ to a $p$-adic representation with the same image. In particular, the ramification properties are unchanged by the lift. Take this $p$-adic representation and view it as a complex representation (i.e. an Artin representation). By Artin's conjecture (which I guess is now a theorem in this 2-dimensional case by Serre's conjecture), there is a weight 1 eigenform $g$ giving rise to this representation. But since the representation is unramified at $q$, we can take $g$ to have level $N/q$, and we're done.

So questions:

0) Does this argument look okay?

1) Is there a more direct argument that doesn't rely on Artin's conjecture to achieve this. (I'm ultimately interested in the Hilbert modular case, and so I don't want to be using relying on Artin's conjecture).

2) Is there a way to handle $p=2,3,5$ even assuming Artin? Is this even true with $p=2$??

Answer by Olivier for Level lowering for weight 1 forms

2012-10-18T18:03:42Z

Everything you wish for is true for modular forms over $\mathbb Q$ even at $p=2$, as it follows from refined forms of Serre's conjecture; here I am assuming of course that $\bar{\rho}$ is absolutely irreducible (I think that you meant to include this explicitly in your set-up, of course otherwise level-lowering can fail). In particular your argument is correct. I don't think there is a significantly more direct general argument incorporating $p=2$ as it is my understanding that it is only with Khare-Wintenberger's proof that the last cases of "Weak Serre implies Refined Serre" were proved.

Over totally real field, this is the whole business of Serre's weight; a topic which has seen exponential development these last 10 years. I am vey far from the most superficial understanding of the current literature but I guess Buzzard-Diamond-Jarvis is a good place to start. Other key players are Toby Gee, Matt Emerton and Florian Herzig. I don't think much is actually known about level-lowering in weight 1, though I could very well be wrong.

Level lowering for weight 1 forms - MathOverflow

Level lowering for weight 1 forms

Answer by Olivier for Level lowering for weight 1 forms