Automorphic-Galois correspondence

The automorphic–Galois correspondence predicts that suitably $L$ -algebraic automorphic representations of a connected reductive group $G$ over a number field $F$ have associated continuous $\ell$ -adic homomorphisms

\rho_{\pi,\iota}: \operatorname{Gal}(\overline F/F) \longrightarrow {}^L G(\overline{\mathbb Q}_\ell),

from the absolute Galois group into the $L$ -group, whose projection to the Galois factor is the canonical one. The representation should be unramified almost everywhere and satisfy local–global compatibility.

This is a general conjecture with many major proved cases, not one theorem in full generality.

Unramified characterization

For almost every finite place $v\nmid\ell$ , the conjugacy class of the Frobenius element $\rho_{\pi,\iota}(\operatorname{Frob}_v)$ should equal the image under $\iota$ of the Satake parameter of $\pi_v$ . Applying an algebraic representation $r:{}^LG\to\operatorname{GL}(V)$ should therefore give matching local Euler factors.

General linear groups

For regular algebraic cuspidal automorphic representations of $\operatorname{GL}_n$ , compatible systems of $n$ -dimensional Galois representations are known in broad settings. The strongest theorem available depends on the base field, regularity, self-duality or polarization, and the place at which compatibility is requested. These hypotheses must not be suppressed when citing a result.

C-algebraic representations

A $C$ -algebraic representation is naturally expected to produce a representation valued in the $C$ -group rather than necessarily in the ordinary $L$ -group. Twisting converts the two normalizations for many general linear groups but not uniformly for all reductive groups.

Not a naive bijection

A Galois representation can determine only an $L$ -packet for a general group, and global multiplicities require additional packet data. Conversely, the compatible $\ell$ -adic realization captures the algebraic part of the automorphic spectrum, whereas a hypothetical global Langlands group is meant to parameterize more general automorphic representations.

References

Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” Conjecture 3.2.1. arXiv.
Laurent Clozel, Michael Harris, and Richard Taylor, “Automorphy for some $\ell$ -adic lifts of automorphic mod $\ell$ Galois representations,” 2008.
Peter Scholze, “On torsion in the cohomology of locally symmetric varieties,” Annals of Mathematics 182 (2015), 945–1066. DOI.