## On the $$p$$-adic cohomology of the Lubin-Tate tower.(English. French summary)Zbl 1419.14031

In this article, the author constructs a functor from the category of admissible $$\bmod p$$ representations of $$\mathrm{GL}_n(F)$$, where $$F/\mathbb{Q}_p$$ is a finite extension, to the category of admissible $$\mathrm{Gal}_F\times D^\times$$-representations, where $$\mathrm{Gal}_F$$ is the absolute Galois group of $$F$$ and $$D/F$$ is the central division algebra of invariant $$1/n$$. His construction provides further evidence for the existence of mod $$p$$ and $$p$$-adic local Langlands and Jacquet-Langlands correspondences. One advantage over a previously established candidate for such correspondences (see [A. Caraiani et al., Camb. J. Math. 4, No. 2, 197–287 (2016; Zbl 1403.11073)]) is that this functor is constructed by purely local means.
To describe the results in slightly more detail, fix a complete and algebraically closed extension $$C \supset F$$. For a smooth admissible representation $$\pi$$ of $$\mathrm{GL}_n(F)$$ on a $$\mathbb{F}_p$$-vector space, the author constructs an étale sheaf $$\mathcal{F}_{\pi}$$ on the adic space $$\mathbb{P}_C^{n-1}$$ using the Lubin-Tate tower and the Gross-Hopkins period map. It is then shown that for all $$i\geq 0$$ the cohomology groups $H_{\text{ét}}^i(\mathbb{P}_C^{n-1},\mathcal{F}_\pi)$ are admissible $$D^\times$$-representations, which carry a continuous action of the absolute Galois group $$\mathrm{Gal}_F$$. They vanish in degree $$i > 2(n-1)$$ and are independent of the choice of the algebraically closed and complete extension $$C$$ of $$F$$.
The proof of this result follows a strategy developed in an earlier work of the author to prove finiteness of $$\mathbb{F}_p$$-cohomology of proper (smooth) rigid analytic spaces, (Zbl 1297.14023). It depends crucially on properness of $$\mathbb{P}^{n-1}_C$$, which is the image of the Gross-Hopkins period map.
The author then proves a local-global compatibility result for $$n=2$$. To describe it, fix a totally real field $$F$$ and a prime $$\mathfrak{p}$$ dividing $$p$$ such that $$F_{\mathfrak{p}}$$ is the $$p$$-adic field considered above. Let $$D_0$$ be a quaternion algebra over $$F$$, which is split at $$\mathfrak{p}$$ and is ramified at all infinite places. Let $$D$$ be the quaternion algebra over $$F$$, which is split at a fixed infinite place $$\infty_F$$, ramified at $$\mathfrak{p}$$ and isomorphic to $$D_0$$ at all other places. Fix a compact open subgroup $$U^{\mathfrak{p}} \subset D_0^\times(\mathbb{A}_{F,f}^{\mathfrak{p}})\cong D^{\times}(\mathbb{A}_{F,f}^{\mathfrak{p}})$$. Let $$\pi = \varinjlim_K S(K U^{\mathfrak{p}},\mathbb{Q}_p/\mathbb{Z}_p)$$ be the direct limit of spaces of algebraic automorphic forms, where $$K$$ runs through the compact open subgroups of $$\mathrm{GL}_2(F_{\mathfrak{p}})\cong D_{0,\mathfrak{p}}$$. Consider the cohomology groups $$H^1(\mathrm{Sh}_{K^\prime U^{\mathfrak{p}},C},\mathbb{Q}_p/\mathbb{Z}_p)$$, of the Shimura curve $$\mathrm{Sh}_{K^\prime U^{\mathfrak{p}}}/F$$ for $$D/F$$, for varying $$K^\prime\subset D_{\mathfrak{p}}^\times$$. The author shows that there is a canonical $$\mathrm{Gal}_{F_{\mathfrak{p}}}\times D^\times_{\mathfrak{p}}$$-equivariant isomorphism $H^1_{\text{ét}}(\mathbb{P}^1_{C},\mathcal{F}_\pi) \cong \varinjlim_{K^\prime} H^1(\mathrm{Sh}_{K^\prime U^{\mathfrak{p}},C},\mathbb{Q}_p/\mathbb{Z}_p).$ He also shows that for $$n=2$$ his functor is compatible with the patching construction of Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin [loc. cit.]. In this context, a new perspective on the patching construction using ultraproducts is explained.
The paper contains an appendix by M. Rapoport on the question of surjectivity of period maps from Rapoport-Zink spaces to partial flag varieties. Two variants of this question are studied. It is first investigated in which situations the period map is surjective on classical points, i.e., for which $$\mathrm{PD}$$-triples the period domain is the whole partial flag variety. It is then proved that surjectivity on all adic points (not just on classical points) occurs essentially only in the Lubin-Tate case.

### MSC:

 14G22 Rigid analytic geometry 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F80 Galois representations 11F85 $$p$$-adic theory, local fields

### Citations:

Zbl 1403.11073; Zbl 1297.14023
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