# Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the residual representation (or make that an assumption) and then infer automorphy of the original representation using deformation-theoretic arguments.

Is it possible to prove automorphy (in the number field context) directly, i.e. without proving the automorphy of the residual representation at any prime?

• If you are looking at complex Galois representations (Artin reps), then yes (Langlands, Tunnell, ...). – Kimball Nov 18 at 13:34

The canonical answer to that question is certainly the world of so called converse theorems, whose basic ideas go back to Hecke's remark that an holomorphic $$L$$-function satisfying a suitable functional equation should be automorphic. In the legendary paper

Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967)

(legendary in particular because it leaves the modularity of elliptic curves over $$\mathbb Q$$ as an exercise for the interested reader), André Weil made this precise and deduced that the $$L$$-function of CM elliptic curves were automorphic. In the language of your question, this shows that Galois representation attached to the torsion points of a CM elliptic curve is automorphic without any consideration of the residual representation.

If you want a conditional, modern incarnation of this principle, consider for instance the following theorem.

Theorem: Let $$\rho:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\longrightarrow\operatorname{GL}_{2}(\mathbb C)$$ be a continuous representation whose $$L$$-function $$L(\rho,s)$$ is holomorphic. Then $$\rho$$ is automorphic.

This theorem (due I believe to Andrew Booker and Muthu Krishnamurty) is proved by a generalization of Weil's converse theorem, and with no consideration of the residual representation. In theory, it applies in situations in which the residual techniques of your question are not at present applicable, namely when $$\rho$$ is even. On the other hand, I don't believe we have any way to check the hypothesis that $$L(\rho,s)$$ is holomorphic except by proving that $$\rho$$ is automorphic, so in that sense the theorem is only a partial success.