Lubin-Tate and Drinfeld bundles
Period rings play a very important role in the study of \(p\)-adic Galois-representations of \(p\)-adic fields. In view of the \(p\)-adic Langlands programme, it is natural to ask whether it is possible to also study representations of \(p\)-adic reductive groups using suitable rings of periods. The paper under review is a first and promising attempt to this question. Let \(K\) be a nonarchimedean local field and denote by \(\mathfrak{o}\) its ring of integers, \(\pi\in\mathfrak{o}\) a fixed uniformizer and let \(h\geq 1\) be an integer. Let \(\check{K}\) be the completion of the maximal unramified extension of \(K\) and \(\check{\mathfrak{o}}\) its valuation ring. Further denote by \(\check{K}_m\) the field obtained by adjoining the \(\pi^m\)-torsion points of a one-dimensional Lubin-Tate formal \(\mathfrak{o}\)-module to \(\check{K}\). For any integer \(m\geq 0\) let \(\mathcal{Y}_m^{(h)}\) be the generic fibre of the formal \(\check{\mathfrak{o}}\)-scheme parametrizing one dimensional formal \(\mathfrak{o}\)-modules of height \(h\) and level structure \(m\) constructed by \textit{V. G. Drinfeld} [Math. USSR, Sb. 23(1974), 561--592 (1976); translation from Mat. Sb., N. Ser. 94(136), 594--627 (1974; Zbl 0321.14014)]. This space admits a commuting left action of the groups \(G_0^{(h)}=\mathrm{GL}_h(\mathfrak{o})\) and \(H_0^{(h)}=\mathfrak{o}_D^\times\) where \(D\) denotes the central division algebra of invariant \(1/h\) over \(K\) and \(\mathfrak{o}_D\) stands for its ring of integers. On the other hand, let \(\mathcal{X}_0^{(h)}:=\Omega_K^{(h)}\times_K \check{K}\) where \(\Omega_K^{(h)}\) is Drinfeld's upper half space of dimension \(h-1\) over \(K\). As shown by \textit{V. G. Drinfeld} [Funct. Anal. Appl. 10, 107--115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29--40 (1976; Zbl 0346.14010)], one may interpret \(\mathcal{X}_0^{(h)}\) as the generic fibre of a formal \(\check{\mathfrak{o}}\)-scheme parametrizing special formal \(\mathfrak{o}_D\)-modules of height \(h^2\). Using this interpretation Drinfeld [op.cit.] constructed a family \((\mathcal{X}_m^{(h)})_{m\geq 0}\) of finite étale Galois coverings of \(\mathcal{X}_0^{(h)}\) on which there also is a commuting left action of the groups \(G_0^{(h)}\) and \(H_0^{(h)}\). Combining the two modular towers of Lubin-Tate and Drinfeld the author defines the space \(\mathcal{Z}_m^{(h)}:=\mathcal{X}_m^{(h)}\times_{\check{K}_m}\mathcal{Y}_m^{(h)}\). For any integer \(m\geq 0\) there are natural morphisms \(p_m:\mathcal{Z}_m^{(h)}\to\mathcal{X}_0^{(h)}\) and \(q_m: \mathcal{Z}_m^{(h)}\to \mathcal{Y}_m^{(h)}\). The question considered in this paper is the following: given a \(G_0^{(h)}\)-equivariant vector bundle \(\mathcal{M}\) on \(\mathcal{X}_0^{(h)}\), under what assumptions are there an integer \(m\geq 0\) and an \(H_0^{(h)}\)-equivariant vector bundle \(\mathcal{N}\) on \(\mathcal{Y}_0^{(h)}\) together with a \(G_0^{(h)}\times H_0^{(h)}\)-equivariant isomorphism \(p_m^*(\mathcal{M})\simeq q_m^*(\mathcal{N})\)? In this case we call \(\mathcal{M}\) Lubin-Tate. Similarly, we may define the notion `Drinfeld' for \(H_0^{(h)}\)-equivariant bundles \(\mathcal{N}\) on \(\mathcal{Y}_0^{(h)}\). The author constructs quasi-inverse equivalences of categories \(\mathbb{D}_{\text{LT}}\) and \(\mathbb{D}_{\text{Dr}}\) between Lubin-Tate and Drinfeld bundles. This is done via a careful understanding of the \(G_0^{(h)}\)- and \(H_0^{(h)}\)-invariant subrings of the global sections of the spaces \(\mathcal{X}_m^{(h)}\) and \(\mathcal{Y}_m^{(h)}\). The nontrivial part of the theory is rather concerned with the construction of nontrivial examples. One of these nontrivial examples is the following: If \(V\) (resp.\ \(W\)) is a finite dimensional smooth representation of \(G_0^{(h)}\) (resp.\ of \(H_0^{(h)}\)) over \(\check{K}\) then the equivariant vector bundle \(\mathcal{M}(V):=\mathcal{O}_{\mathcal{X}_0^{(h)}}\otimes_{\check{K}}V\) (resp.\ \(\mathcal{N}(W):=\mathcal{O}_{\mathcal{Y}_0^{(h)}}\otimes_{\check{K}}W\)) is Lubin-Tate (resp.\ Drinfeld). In order to study objects that are equivariant under the full groups \(G^{(h)}=\mathrm{GL}_h(K)\) and \(H^{(h)}=D^*\), the author considers the Rapoport-Zink spaces \(\underline{\mathcal{X}}_m^{(h)}\) and \(\underline{\mathcal{Y}}_m^{(h)}\), as well as the corresponding period spaces \(\mathcal{X}_0^{(h)}\) and \(\mathbb{P}_{\check{K}}^{h-1}\) and establishes a similar theory to the above mentioned.
scientific article; zbMATH DE number 5955297
scientific article