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Langlands correspondence for GL2\mathrm{GL}_2 over imaginary quadratic fields

Let KK be an imaginary quadratic field, and let π\pi be a cuspidal automorphic representation of GL2(AK)\mathrm{GL}_2(\mathbb A_K) as in Taylor's theorem, but without assuming that its central character is invariant under the nontrivial automorphism of KK or that the integer kk in its Langlands parameter is even. For each prime λ\lambda of the coefficient field, let

ρλ:Gal⁡(K‾/K)→GL2(Eλ)\rho_{\lambda}:\operatorname{Gal}(\overline K/K)\to \mathrm{GL}_2(E_{\lambda})

be the associated Galois representation.

Langlands correspondence for GL2/K\mathrm{GL}_2/K. The representation ρλ\rho_{\lambda} exists as above, is potentially semistable at every place vv, and its associated representation of the Weil–Deligne group satisfies the local Langlands correspondence with the representation πv\pi_v.

This is the conjectural extension of Taylor's construction of Galois representations to general cuspidal automorphic representations in this setting. It is intended to provide finer control at ramified places and at primes dividing the residue characteristic; the source gives no resolution.

Sources & referencesView supporting material

Primary source

Frank Calegari and Nathan M Dunfield, “Automorphic forms and rational homology 3–spheres”, arXiv:math/0508271 (2009).

automorphic-formsgalois-representationslanglands-correspondence

Progress summary

Refreshed
Open

The unrestricted version remains open: earlier work handles important special cases, but no complete proof or counterexample was found.

The conjecture extends Taylor’s construction of Galois representations from suitable cuspidal representations of GL2(AK)\mathrm{GL}_2(\mathbb A_K) to arbitrary ones, including non-invariant central character and odd parameter weight. It also predicts potential semistability and local Langlands compatibility.

Known results

  • Taylor’s theorem constructs compatible systems under the hypothesis ωπc=ωπ\omega_\pi^c=\omega_\pi.
  • Berger and Harcos construct representations under the same central-character hypothesis, with Hecke–Frobenius compatibility outside an explicit finite set.
  • Taylor and collaborators prove local-global compatibility for regular algebraic, essentially conjugate-self-dual representations over CM fields, which remains a restricted setting.

Current status (as of August 2026): The correspondence is established only in restricted cases, notably when ωπc=ωπ\omega_\pi^c=\omega_\pi or under essential conjugate self-duality; the unrestricted GL2/K\mathrm{GL}_2/K statement remains open.

Sources

Solutions 0

No solutions have been posted yet.