Shimura varieties

Shimura varieties
Name	Shimura varieties
Established	1960s
Field	Number theory; Algebraic geometry; Representation theory
Notable	Goro Shimura; Yasutaka Ihara; Jean-Pierre Serre

Shimura varieties are higher-dimensional algebraic varieties that arise as moduli spaces for certain types of Hodge structures and abelian varieties with extra structure. They bridge arithmetic geometry, automorphic representations, and Langlands reciprocity, connecting the work of Goro Shimura, Yutaka Taniyama, Yasutaka Ihara, Jean-Pierre Serre, Pierre Deligne, and Robert Langlands. Shimura varieties play central roles in the proofs of cases of the Taniyama–Shimura–Modularity theorem and in the formulation of the Langlands program.

Introduction

Shimura varieties were introduced in the 1960s by Goro Shimura and Yutaka Taniyama and developed extensively by Pierre Deligne and James Milne. They generalize classical moduli spaces like the modular curve and the Siegel modular variety, and relate to Eichler–Shimura phenomena, Hecke operators, and adelic techniques. Over complex numbers, Shimura varieties are quotients of hermitian symmetric domains by arithmetic subgroups such as congruence subgroups of GL(2), GSp(2g), or more general reductive groups studied by Armand Borel and Harish-Chandra. Their study involves tools from Hodge theory, etale cohomology, and arithmetic geometry.

Definitions and Basic Properties

A Shimura datum is a pair (G, X) where G is a connected reductive algebraic group over Q and X is a G(R)-conjugacy class of homomorphisms from the Deligne torus to G_R satisfying Deligne's axioms; this formalism was systematized by Pierre Deligne and used by James Milne and G. Laumon. For a compact open subgroup K of G(A_f) one forms the double quotient G(Q)\X×G(A_f)/K which, via Baily–Borel and the theory of arithmetic subgroups developed by Armand Borel and Harish-Chandra, has the structure of a complex algebraic variety. Important structural properties include the action of Hecke algebra elements, canonical models over reflex fields defined by Deligne, and the existence of integral models studied by M. Rapoport and T. Zink.

Examples and Special Cases

Classical examples include modular curves associated to GL(2), which parameterize elliptic curves with level structure as in work of Hecke. The Siegel modular varieties attached to GSp(2g) parametrize principally polarized abelian varieties and link to Faltings's work on the Mordell conjecture and the Torelli theorem. Hilbert modular varieties come from restriction of scalars from a totally real field as in constructions by Ernst Kummer successors; Picard modular surfaces arise from unitary groups over CM fields investigated by Emil Artin and Jacques Tits. Certain orthogonal Shimura varieties relate to moduli of K3 surfaces studied by Shigefumi Mori and Phillip Griffiths.

Canonical Models and Arithmetic Aspects

Deligne defined reflex fields and canonical models for Shimura varieties, enabling interpretation over number fields and making sense of Galois actions on special points or CM points as studied in the André–Oort conjecture context, addressed by contributions from Jonathan Pila and Christopher Daw. The theory of complex multiplication for abelian varieties links to the reciprocity laws of Class field theory and to explicit results of Yves Henniart and Jean-Pierre Serre. Integral models at primes of bad reduction were developed by M. Rapoport, T. Zink, Mark Kisin, and George Pappas, connecting to p-adic Hodge theory and Fontaine's theory.

Cohomology and Automorphic Forms

The l-adic and Betti cohomology of Shimura varieties realize instances of the Langlands correspondence: automorphic representations of G contribute to the cohomology via the work of Robert Langlands, James Arthur, Mark Kisin, Michael Harris, and Richard Taylor. The Eichler–Shimura isomorphism and its generalizations link modular forms to the middle-degree cohomology, while contributions of Gerd Faltings and Jean-Marc Fontaine connect p-adic Hodge theoretic comparisons. The trace formula of James Arthur and the stabilization techniques of Robert Kottwitz play roles in counting points over finite fields and relating Frobenius eigenvalues to automorphic L-functions investigated by Harris–Taylor.

Compactifications and Geometry

Shimura varieties admit several compactifications: Baily–Borel (minimal) compactifications introduced by Walter Baily and Armand Borel, toroidal compactifications developed by Caraiani-era predecessors and by Faltings–Chai in the Siegel case, and various minimal and log compactifications studied by Kazuya Kato and Michael Rapoport. Geometric properties such as the existence of ample automorphic line bundles, the Kodaira dimension computations by Gerd Faltings and collaborators, and stratifications like the Newton and Ekedahl–Oort strata were analyzed by M. Rapoport, T. Zink, Eyal Z. Goren, and Ben Moonen.

Applications and Open Problems

Shimura varieties are central in proofs of modularity lifting theorems by Richard Taylor and Andrew Wiles style methods, in the proof of the Sato–Tate conjecture for elliptic curves via work of Richard Taylor and Christophe Breuil, and in the study of special values of L-functions pursued by Don Zagier and Pierre Deligne. Open problems include the full Langlands reciprocity for general reductive groups advocated by Robert Langlands, the general André–Oort and Zilber–Pink conjectures addressed by Jonathan Pila and Umberto Zannier, and the construction of good integral models in full generality pursued by Mark Kisin and M. Rapoport. Other active directions involve p-adic local Langlands as developed by Peter Scholze and Matthew Emerton and interactions with motivic conjectures of Grothendieck.

Category:Algebraic geometry