nLab moduli stack of L-parameters

Idea

The moduli stack of L-parameters is an algebraic moduli stack that parametrizes L-parameters, which generalize the Weil-Deligne representations for the local Langlands correspondence when the reductive group is not the general linear group $GL_n$.

In the “classical” $\ell$-adic case of the local Langlands correspondence, this moduli stack was constructed independently by Dat-Helm-Kurinczuk-Moss (DHKM20), Fargues-Scholze (FarguesScholze21), and Zhu (Zhu20). For the p-adic local Langlands correspondence, the relevant stack has to be the moduli stack of etale $(\varphi,\Gamma)$-modules (also known as the Emerton-Gee stack), constructed by Emerton-Gee (EmertonGee19).

The corresponding notion in the geometric Langlands correspondence is the moduli stack of local systems (recall – e.g. from Deligne 1970 – that local systems on a curve (or any space) give linear representations of its fundamental group, hence are akin to Galois representations).

References

• Allan J. Silberger, Ernst-Wilhelm Zink, Section 3 of: Langlands Classification for L-Parameters, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 (arXiv:1407.6494, doi:10.1016/j.jalgebra.2018.06.012)

• Ernst-Wilhelm Zink, with Allan J. Silberger, Langlands classification for L-parameters (pdf)

• Jean Francois Dat, David Helm, Robert Kurinczuk, and Gilbert Moss, Moduli of Langlands Parameters (arXiv:2009.06708)

• Xinwen Zhu, Coherent sheaves on the stack of Langlands parameters (arXiv:2008.02998)

• Laurent Fargues, Peter Scholze, Geometrization of the local Langlands correspondence (arXiv:2102.13459)

• Matthew Emerton, Toby Gee, Moduli stacks of etale (phi,Gamma) modules and the existence of crystalline lifts (arXiv:1908.07185)