Cerednik–Drinfeld theory
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| Cerednik–Drinfeld theory | |
|---|---|
| Name | Cerednik–Drinfeld theory |
| Field | Number theory, Arithmetic geometry |
| Introduced | 1970s–1980s |
| Contributors | Igor Shafarevich, Vladimir Drinfeld, Ilya Cerednik, Jean-Pierre Serre, Gérard Laumon |
Cerednik–Drinfeld theory is a body of results establishing p-adic uniformization of certain algebraic curves and Shimura varieties, giving analytic descriptions of reductions and moduli in the non-archimedean setting. The theory connects rigid analytic spaces, formal schemes, and p-adic groups to arithmetic of quaternion algebras, modular curves, and Shimura varieties associated to reductive groups like GL(2), GSp(4), and inner forms. It lies at the intersection of work by John Tate, Pierre Deligne, Jacques Tits, Alexander Grothendieck, and later developments by Richard Taylor and Andrew Wiles.
Introduction
Cerednik–Drinfeld results give a p-adic analytic uniformization for curves and higher-dimensional varieties related to Shimura varieties, describing their structure in terms of quotients of p-adic symmetric domains by arithmetic subgroups of p-adic reductive groups such as PGL(2), GU(1,1), and groups attached to quaternion algebras. The theory ties reductions modulo p studied by Jean-Marc Fontaine, Gerd Faltings, and Mazur to rigid analytic and formal models influenced by Michel Raynaud, Kazuya Kato, and Ofer Gabber. It provides tools used in the proofs of modularity and local-global compatibility in the work of Taylor–Wiles, Breuil, Emerton, and others.
Background and preliminaries
Key prerequisites include the theory of p-adic numbers, p-adic Hodge theory, and the theory of formal schemes and rigid analytic spaces developed by John Tate and refined by Raynaud and Bosch-Lütkebohmert. Understanding quaternion algebras over number fields as in the work of Adrien-Marie Legendre's successors and the classification by Ernst Witt and Helmut Hasse is essential, as are the arithmetic of modular curves studied by Hecke and Atkin–Lehner and the local structure of reductive groups from Bruhat–Tits theory by François Bruhat and Jacques Tits. The role of level structures involves congruence subgroups such as those considered by Franz Lemmermeyer and the moduli interpretation connects to stacks and deformation theory from Alexander Grothendieck and Pierre Deligne.
Statement of Cerednik–Drinfeld theorem
Roughly, for a quaternion algebra over a number field ramified at a finite place p and at infinity in a prescribed way, the theorem asserts that the p-adic completion of the associated Shimura curve is isomorphic to the quotient of a p-adic symmetric domain (Drinfeld upper half plane) by a discrete subgroup of PGL(2,Q_p) or an inner form, giving an explicit p-adic uniformization. The precise formulation builds on the cohomological methods of Andre Weil and moduli interpretations akin to those of Igor Shafarevich and Serre-Tate and uses comparison isomorphisms reminiscent of Fontaine's rings and the crystalline techniques of Gerd Faltings. Variants treat bad reduction and link to the component groups studied by Barry Mazur and David Mumford.
Construction of p-adic uniformization
The construction uses the Drinfeld upper half plane introduced by Vladimir Drinfeld and its coverings by p-adic analytic spaces studied by Christophe Breuil and Guy Henniart. One builds a formal model via Raynaud's theory and computes the action of an arithmetic subgroup arising from an order in a quaternion algebra, with input from Bruhat–Tits buildings and the reduction theory of Tamagawa and Weil; this yields an explicit rigid-analytic quotient. Cohomological control employs techniques from Évariste Galois-style descent, Jean-Pierre Serre's local factors, and the p-adic comparison theorems of Fontaine and Faltings. The analysis of degenerations uses special fibers related to models studied by Mumford and deformation spaces linked to Mazur and Grothendieck.
Applications and consequences
Consequences include descriptions of the bad reduction of Shimura curves used in arithmetic of modular forms and the construction of local Langlands correspondences paralleling results of Harris–Taylor and Henniart. The theory underpins computations of local components of automorphic representations in the work of Jacquet–Langlands and informs the proof strategies in Wiles and Taylor-type modularity lifting arguments. It influences explicit class field theory for CM fields and the study of rational points via Chabauty-type methods developed by Coleman and Kim. It also connects to non-abelian generalizations in the research programs of Langlands and to geometric approaches pursued by Laumon, Rapoport, and Zink.
Examples and explicit cases
Prototype examples include the p-adic uniformization of classical modular curves at primes dividing the discriminant of a related quaternion algebra, illustrated in computations by Eichler and Shimura. Explicit uniformizations for Shimura curves attached to indefinite quaternion algebras over Q were developed in the original works and later generalized to Hilbert and unitary cases treated by Hulek, Rapoport–Zink, and Kudla. Concrete cases connect to the moduli of elliptic curves with level structure as studied by Katz–Mazur and to degenerations appearing in computations by Besser, Berthelot, and Fargues. These instances furnish explicit descriptions of special fibers relevant to the arithmetic of Heegner points and the Gross–Zagier formula involving Gross and Zagier.