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| Rapoport–Zink | |
|---|---|
| Name | Rapoport–Zink |
| Discipline | Mathematics |
| Subdiscipline | Arithmetic geometry, Algebraic geometry, Number theory |
| Introduced | 1990s |
| Creators | Michael Rapoport, Thomas Zink |
Rapoport–Zink. Rapoport–Zink are foundational constructions in arithmetic geometry introduced by Michael Rapoport and Thomas Zink that provide formal moduli spaces parametrizing p-divisible groups with additional structures related to reductive groups such as GL_n, GSp_2n, and U(n). They connect the work of researchers like Pierre Deligne, Jean-Marc Fontaine, Gerard Laumon, Kazuya Kato, and Gerd Faltings to problems studied by groups including Institute for Advanced Study, Max Planck Institute for Mathematics, and universities such as Harvard University and University of Bonn. Rapoport–Zink constructions serve as local analogues of global moduli spaces associated to Shimura varieties, complementing theories by Richard Taylor, Andrew Wiles, Robert Langlands, and James Arthur.
Rapoport–Zink objects are formal schemes or adic spaces over complete discrete valuation rings such as Witt vectors of perfect fields like F_p^alg that parametrize deformations of a fixed p-divisible group with quasi-isogenies and level structures, relating to linear algebraic groups like GSp_4 and Res_{F/\Q}GL_n. They are defined using data including a reductive group over Q_p, a conjugacy class of cocharacters à la Deligne and Milne, and a basic isocrystal from the classification of Dieudonné modules by Jean-Pierre Serre and John Tate. Their local structure mirrors objects studied by Alexander Grothendieck and Luc Illusie in crystalline cohomology and displays symmetry under actions of groups such as J_b(Q_p), Iwahori subgroup, and Weil group representations.
The genesis traces to moduli problems pioneered by Igusa and to the study of formal moduli by Grothendieck in the 1960s, followed by advances in p-adic Hodge theory by Fontaine in the 1980s and the formulation of Shimura varieties by Shimura and Taniyama. Rapoport and Zink synthesized these strands in the 1990s, building on work by Nicholas Katz, Ofer Gabber, Jean-Pierre Serre, and contemporaries including Peter Scholze and Mark Kisin, and responding to problems raised in the Langlands program by Robert Langlands and applications such as the proof of modularity by Wiles. Subsequent developments involved collaborations with scholars from Princeton University, University of Cambridge, and École Normale Supérieure.
Construction begins with a framing object: a p-divisible group with additional endomorphisms by rings like O_F for a local field F and a polarization compatible with groups like GSp_{2n}, together with an isocrystal classified by Kottwitz invariants and Newton polygon data studied by Uffe Haagerup and Peter Scholze. The formal moduli space is a formal scheme locally formally of finite type over Spf of Witt vectors, often endowed with a period morphism to flag varieties such as Grassmannian or to period domains appearing in work by Carlos Simpson and Rapoport. These spaces carry actions of groups like J_b and Hecke correspondences studied by Ilya Piatetski-Shapiro and admit stratifications reminiscent of Ekedahl–Oort strata and Newton strata analyzed by Francesc Fité and C. D. Skinner. Their deformation theory uses techniques from Faltings and Kisin on crystalline and semistable representations.
Notable instances include the Lubin–Tate case linked to John Tate and local class field theory, where the Rapoport–Zink space is isomorphic to the deformation space of a one-dimensional formal group and relates to the local Langlands correspondence studied by Colmez and Henniart. The Drinfeld case connects to work by Vladimir Drinfeld on moduli of shtukas and to Kazhdan–Lusztig theory via p-adic period domains. Unitary Rapoport–Zink spaces tied to U(1,n-1) feature in investigations by Wei Zhang and Rapoport on arithmetic intersections, while symplectic cases for GSp_{2n} underpin applications pursued by Laurent Fargues and Peter Scholze. Special points relate to CM types examined by Shimura and Yoshida.
Rapoport–Zink spaces serve as local models for the integral and p-adic uniformization of Shimura varieties studied by Deligne, Carayol, and Kottwitz. They appear in p-adic uniformization theorems connecting global moduli problems represented by Shimura varieties to local deformation spaces at primes studied by Rapoport and Zink, with links to global automorphic forms investigated by Jacquet and Langlands. These spaces give local contributions to the study of arithmetic compactifications considered by Faltings and Chai, and feed into the construction of local models by Pappas and Rapoport that mirror singularities appearing in the integral models of Shimura varieties.
Applications include comparisons between l-adic and p-adic cohomology groups used in proofs involving Langlands correspondence instances by Harris and Taylor, the study of local factors of automorphic L-functions considered by Godement and Jacquet, and contributions to arithmetic intersection conjectures as in work by Kudla and Rapoport. They facilitate explicit computations of local terms in trace formulae employed by Arthur and underpin constructions in p-adic Hodge theory central to research by Scholze and Fargues–Fontaine.
Open problems include the fine structure of cohomology of Rapoport–Zink spaces in relation to the local Langlands correspondence conjectures by Harris–Taylor and refinements posed by Kottwitz and Rapoport, the determination of non-basic loci connected to conjectures by Mantovan, and extensions to mixed characteristic analogues investigated by Scholze and W. Kim. Recent progress includes advances in v-sheaf and diamond frameworks by Peter Scholze linking to the Fargues–Fontaine curve studied by Laurent Fargues and applications to categorical Langlands programs pursued by Edward Frenkel and collaborators. Further work at institutions like IHÉS, MSRI, and departments across Princeton University and Cambridge University continues to push boundaries.