Let \(F\) be a local field: if \(\text{ char}(F)=0\) then \(F\) is a finite extension of \({\mathbb Q}_p\), but if \(\text{char}(F)=p>0\) then \(F\cong{\mathbb F}_q((\pi))\) for a finite field \({\mathbb F}_q\). Suppose for the moment that \(F={\mathbb Q}\) and let \(W_F\) denote its Weil group. Let \(n\geq2 \) and let \(D\) be a division algebra of invariant \(\frac{1}{n}\) over \(F\). The Lubin-Tate tower and the Drinfeld tower are certain infinite towers of \(F\)-rigid analytic spaces endowed with actions of \(\text{GL}_n(F)\times D^{\times}\times W_F\) which are defined by moduli problems parametrizing certain \(p\)-divisible groups with additional structures. Similarly for any \(F\). For \(\text{char}(F)=0\) these towers are \(p\)-adic local analogs of Shimura varieties (which in their turn are towers of algebraic varieties over a number field, e.g. modular curves). They are special cases of moduli spaces for \(p\)-divisible groups as defined in full generality by M. Rapoport and T. Zink [“Period spaces for \(p\)-divisible groups.” Ann. Math. Stud. 141. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)]. For \(\text{char}(F)=p>0\) these towers are local analogs of moduli spaces for Shtukas. The moduli problems and the actions of the group \(\text{GL}_n(F)\times D^{\times}\) are in a certain sense perpendicular: while \(\text{GL}_n(F)\) acts ‘vertically’ (i.e. by Hecke correspondences) on the Lubin-Tate tower and ‘horizontally’ (i.e. stabilizing each layer) on the Drinfeld tower, \(D^{\times}\) acts horizontally on the Lubin-Tate tower and vertically on the Drinfeld tower. It was expected for a long time, and was recently established for the Lubin-Tate tower by M. Harris and R. Taylor [“The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich.” Ann. Math. Stud. 151, Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)], that the \(\ell\)-adic \((\ell\neq p)\) étale cohomology of either of these towers realizes the local Langlands correspondence. So it seemed not unreasonable to also expect some geometric relation between the two towers themselves, but even a clean formulation of this expectation was missing. Then G. Faltings [in: Algebraic number theory and algebraic geometry. Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 300, 115–129 (2002; Zbl 1062.14059)] indeed constructed an isomorphism between these towers ‘at infinite level’ (implying, due to the result of Harris and Taylor, that the local Langlands correspondence is also realized in the cohomology of the Drinfeld tower). The aim of this book is to give a detailed proof of this result, as well as further complements and applications. The book splits up in two parts. The first part (p. 3–326), written by L. Fargues, works out Faltings’ method in the case \(\text{char}(F)=0\). The second part (p. 327–406), written by A. Genestier and V. Lafforgue, presents an alternative approach which works only in the case \(\text{char}(F)=p>0\).
Let again for simplicity \(F={\mathbb Q}\). The base space of the Lubin-Tate tower is the \(n-1\)-dimensional unit disc \[ {\mathfrak X}_{\eta}=\{(x_1,\ldots,x_{n-1})\in{\mathbb C}_p^{n-1};| x_i| <1\}. \] This is the ‘generic fibre’ of the formal scheme \({\mathfrak X}\) which parametrizes deformations of a formal group \({\mathbb H}\) of dimension \(1\) and height \(n\) over \(\overline{\mathbb F}_p\). This \({\mathbb H}\) is uniquely determined up to isomorphism. For example, if \(n=2\) then \({\mathbb H}\) is isomorphic to the formal completion at the origin of a supersingular elliptic curve over \(\overline{\mathbb F}_p\). Let \({\mathcal O}_D\) denote the maximal order in \(D\). Then \[ {\mathcal O}_D^{\times}\cong\operatorname{Aut}({\mathbb H}) \] acts on \({\mathfrak X}\), hence on \({\mathfrak X}_{\eta}\). Let \(H\) denote the universal deformation of \({\mathfrak X}\) and \(H[p^k]\) its \(p^k\)-torsion points. The generic fibre \(H[p^k]_{\eta}\) is a finite étale group over \({\mathfrak X}_{\eta}\), a rigid étale local system in \({\mathbb Z}/p^k{\mathbb Z}\)-modules locally free of rank \(n\). Partial trivialisations of this local system then give rise to the Lubin-Tate tower \[ (\mathcal {LT}_K)_{K\subset\text{GL}_n({\mathbb Z}_p)} \] of finite étale coverings of \(\mathcal{LT}_{\text{GL}_n({\mathbb Z}_p)}=\coprod_{\mathbb Z}{\mathfrak X}_{\eta}\), indexed by the open subgroups of \(\text{GL}_n({\mathbb Z}_p)\). For example, the principal congruence subgroups \(K=\text{Id}+p^kM_n({\mathbb Z}_p)\) correspond to the trivializations of \(H[p^k]_{\eta}\). The vertical action of \(\text{GL}_n(F)\) by Hecke correspondences permutes the various subgroups \(K\subset \text{GL}_n({\mathbb Z}_p)\). The base space for the Drinfeld tower is Drinfeld’s space \[ \Omega({\mathbb C}_p)={\mathbb P}^{n-1}({\mathbb C}_p)-\bigcup_{H\in \check{\mathbb P}^{n-1}(F)}H({\mathbb C}_p) \] endowed with its natural action by \(\text{GL}_n(F)\). For example, taking \(n=2\) we get the \(p\)-adic upper half plane \(\Omega({\mathbb C}_p)={\mathbb C}_p-{\mathbb Q}_p\). There is a semistable formal model \(\widehat{\Omega}\) of \(\Omega\) whose irreducible components and their intersections are parametrized by the Bruhat Tits building of \(\text{PGL}_n(F)\). This formal scheme \(\widehat{\Omega}\) parametrizes deformations up to quasiisogeny of certain formal groups of dimension \(n\) and height \(n^2\), endowed with an action by \({\mathcal O}_D\). Over \(\overline{\mathbb F}_p\) there is up to isogeny a unique such formal group \({\mathbb G}\); it satisfies \[ \text{ End}_{{\mathcal O}_D}({\mathbb G})_{\mathbb Q}=M_n(F), \] hence a modular interpretation of the action of \(\text{GL}_n(F)\) on \(\widehat{\Omega}\). Just as in the Lubin-Tate case one now defines a tower \[ (\mathcal{D}r_K)_{K\subset{\mathcal O}_D^{\times}} \] over \(\mathcal{D}r_{{\mathcal O}_D^{\times}}=\coprod_{\mathbb Z}\Omega\): namely, \(\mathcal{D}r_K\) is defined by asking that the monodromy representation of \(\pi_1(\Omega)\) on the \(p\)-adic Tate module of the universal \(p\)-divisible group – this Tate module is free of rank on over \({\mathcal O}_D\) – lives in \(K\). On the Drinfeld side, the rigid spaces \(\mathcal{D}r_K\) have natural underlying \(p\)-adic formal schemes \(\widehat{\mathcal{D}r}_K\) and one may form \[ \widehat{\mathcal{D}r}_{\infty}=\lim_{\leftarrow}\widehat{\mathcal{D}r}_K \] in the category of \(p\)-adic formal schemes. On the Lubin-Tate side there are no natural underlying \(p\)-adic formal schemes availabe. Nevertheless, the central theorem proved by Faltings is:
Theorem 1. (Faltings) There exists a ‘\(p\)-adification’ \(\widetilde{\widehat{\mathcal {LT}}}_{\infty}\) of the Lubin-Tate tower ‘at infinite level’ and an isomorphism \[ \widetilde{\widehat{\mathcal {LT}}}_{\infty}\cong \widetilde{\widehat{\mathcal{D}r}}_{\infty} \] where \(\widetilde{\widehat{\mathcal{D}r}}_{\infty}\) is a certain admissible formal blowing up of \({\widehat{\mathcal{D}r}}_{\infty}\). This isomorphism is equivariant for the action of \(\text{GL}_n(F)\times D^{\times}\) (up to twisting the action of \(\text{GL}_n(F)\) by its obvious involution).
The principal application lies in the following theorem:
Theorem 2. (a) There exists an equivalence of topoi \[ \lim_{\leftarrow\atop K\subset \text{GL}_n(\mathcal {O}_F)}(\mathcal {LT}_K)^{\tilde{}}_{\text{rig-ét}}\cong\lim_{\leftarrow\atop K\subset\mathcal {O}_D^{\times}}(\mathcal{D}r_K)_{\text{rig-ét}}^{\tilde{}} \] compatible with the action of \(\text{GL}_n(F)\times D^{\times}\).
(b) Let \(\Lambda\) be a torsion ring. There is for any \(q\geq0\) a \(\text{GL}_n(F)\times D^{\times}\times W_F\)-equivariant isomorphism \[ \lim_{\rightarrow\atop K\subset \text{GL}_n(\mathcal {O}_F)}H^q_c(\mathcal {LT}_K\widehat{\otimes}{\mathbb C}_p,\Lambda)\cong \lim_{\rightarrow\atop K\subset \mathcal {O}_D^{\times}}H^q_c(\mathcal{D}r_K\widehat{\otimes}{\mathbb C}_p,\Lambda). \]
In fact there is a more precise version in the derived category of equivariant \(\Lambda\)-modules. Moreover, there is then a ‘Jaquet-Langlands’ equivalence of topoi between rig-étale sheaves with smooth \(D^{\times}\)-action on the Gross-Hopkins period space \({\mathbb P}^{n-1}\) (to which the Lubin-Tate tower maps by a period map), and the analogous sheaves on \(\Omega\).
A major problem one faces when trying to make sense of an isomorphism ‘at infinity level’ between the two towers (as stated in Theorem 1) is that the inverse limits \(\lim_{\leftarrow}\mathcal{D}r_K\) resp. \(\lim_{\leftarrow}\mathcal {LT}_K\) do not exist in the classical theory of rigid spaces. From a foundational point of view one of the merits of the present book is that it profoundly lays the grounds for treating such limiting processes in great generality. Partly in preparation for this, but in fact with a much broader scope one finds in detail many basic facts on the approach to rigid spaces via formal schemes (e.g. blowing ups, normalizations, limits, ...). Note that at present, while there are several textbooks available on rigid spaces, there seems to be no one availabe particularly devoted to this topic. The book aims to be self contained, proofs are given in full detail. The inputs needed from Dieudonné theory (crystals), Rapoport-Zink period spaces and period maps, (rig)-étale sheaf- and topos theory, equivariant cohomology etc. are carefully presented in various appendices to the main text, complemented by various useful digressions. Thus, the reader finds a wealth of valuable material on many topics related to the proper subject of this book. To anyone working on \(p\)-adic period spaces, \(p\)-adic moduli spaces or geometric aspects on the local Langlands program this book can highly be recommended.