nLab p-adic local Langlands correspondence




The pp-adic local Langlands correspondence is a version of the local Langlands correspondence where the representations are over a pp-adic field, i.e. on one side we have certain EE-representations of G(F)G(F) (for GG some reductive group), and on the other we have certain EE-representations of Gal(F¯/F)\mathrm{Gal}(\overline{F}/F), where EE and FF are both finite extensions of p\mathbb{Q}_{p}. It should be compatible with the mod p local Langlands correspondence. The only case that is known is when G=GL 2G=\mathrm{GL}_{2} and F= pF=\mathbb{Q}_{p}.

An approach to realizing one direction of the p-adic local Langlands correspondence using patching was developed in #CEGGPS2013. An approach going in the other direction was developed in #Scholze2015. These two constructions are compatible with each other. Scholze’s approach was further developed in #HansenMann2022.

Scholze’s p-adic Jacquet-Langlands functor

We describe the construction in #Scholze2015. Let FF be a finite extension of p\mathbb{Q}_{p} and let C=F¯^C=\widehat{\overline{F}}. Let \mathcal{M}_{\infty} be the infinite level Lubin-Tate space over CC. It is a perfectoid space equipped with commuting actions of GL n(F)\GL_{n}(F) and D ×D^{\times}, where D/FD/F is the central simple algebra of invariant 1/n1/n. There is a Gross-Hopkins period map π GH: C n1\pi_{\mathrm{GH}}:\mathcal{M}_{\infty}\to\mathbb{P}_{C}^{n-1}.

Let π\pi be a smooth 𝔽 p\mathbb{F}_{p}-representation of GL n(F)\GL_{n}(F). We define a sheaf π\mathcal{F}_{\pi} on C n1\mathbb{P}_{C}^{n-1} as follows. Given an etale map U C n1U\to\mathbb{P}_{C}^{n-1}, define

π=Map cont,GL n(F)×D ×(|U× C n1 |,π).\mathcal{F}_{\pi}=\mathrm{Map}_{\mathrm{cont},\GL_{n}(F)\times D^{\times}}(\vert U\times_{\mathbb{P}_{C}^{n-1}} \mathcal{M}_{\infty}\vert,\pi).

Scholze’s p-adic Jacquet-Langlands functor 𝒥 i\mathcal{J}^{i} is now given by the ii-th etale cohomology of the sheaf π\mathcal{F}_{\pi}:

𝒥 i(π)=H i( C n1, π)\mathcal{J}^{i}(\pi)=H^{i}(\mathbb{P}_{C}^{n-1},\mathcal{F}_{\pi})

The representations 𝒥 i(π)\mathcal{J}^{i}(\pi) are smooth 𝔽 p\mathbb{F}_{p}-representations of D ×D^{\times} and also carry an action of the Weil group W FW_{F}.


Last revised on July 30, 2023 at 20:13:21. See the history of this page for a list of all contributions to it.