Shimura curve

Shimura curve
Name	Shimura curve
Field	Number theory, Algebraic geometry
Introduced	1950s
Related	Shimura variety, Modular curve, Quaternion algebra, Abelian variety

Shimura curve Shimura curves are algebraic curves arising as one-dimensional cases of Shimura variety associated to quaternion algebras and certain arithmetic groups. They generalize modular curve constructions connected to elliptic curves, automorphic forms, and Galois representations, and they play central roles in results of Goro Shimura, Yutaka Taniyama, Andrew Wiles, and Pierre Deligne. Shimura curves link classical objects such as Hilbert modular surfaces, Siegel modular varietys, and Drinfeld modules to modern approaches in Langlands program, Iwasawa theory, and arithmetic geometry.

Introduction

Shimura curves were introduced by Goro Shimura in the context of the study of complex multiplication and Hecke operators, extending the paradigm provided by Modular curve X_0(N) and Modular form theory of Ernst Hecke and Atkin–Lehner theory. They are typically associated with indefinite quaternion algebras over totally real fields studied by Richard Taylor and Jacques Tits, and they appear in the work of John Milnor, Jean-Pierre Serre, and Nicholas Katz on arithmetic compactification and cohomology. Shimura curves serve as test-cases for conjectures of Robert Langlands, Jean-Pierre Serre, and Barry Mazur concerning modularity, level-raising, and rational points.

Definition and Construction

A Shimura curve is constructed from a reductive algebraic group G over Q with an associated hermitian symmetric domain D and a compact open subgroup K of G(A_f), following the Shimura datum formalism of Pierre Deligne and James Milne. In the quaternionic case one takes G to be the multiplicative group of a quaternion algebra B over a totally real field F, split at exactly one infinite place, and K corresponding to an arithmetic subgroup such as an Eichler order studied by Eichler and Hiroshi Saito. The complex points of the Shimura curve are quotients of the upper half-plane by discrete arithmetic subgroups modeled on Fuchsian group theory of Henri Poincaré and Felix Klein, while the algebraic model over a number field follows Shimura reciprocity and canonical models developed by André Weil and Taniyama–Shimura conjecture contributors including Gerhard Frey.

Complex and Arithmetic Properties

As Riemann surfaces, Shimura curves admit uniformizations by Poincaré half-plane quotients arising from arithmetic Fuchsian groups studied by Harald Bohr and Atle Selberg; their geodesic and spectral properties relate to the Selberg trace formula and results of Peter Sarnak and Don Zagier. Arithmetic properties involve reduction mod p analyzed by Jean-Marc Fontaine and Barry Mazur through p-adic Hodge theory, Fontaine–Mazur conjecture aspects, and potential good reduction criteria due to Serre–Tate theory and Grothendieck techniques. Hecke correspondences on Shimura curves produce endomorphisms of their Jacobians studied by Goro Shimura, Jacobian variety theory of André Weil, and the work of Ribet on level-lowering.

Examples and Notable Families

Classical examples include curves associated with quaternion algebras over Q ramified at prescribed sets of primes, such as curves arising in the work of Shimura 1958 and families used by Ken Ribet and Andrew Wiles in proofs of modularity lifting. Notable families comprise curves over real quadratic fields explored by Erez Lapid and Dorian Goldfeld, and special cases linked to Gross–Zagier formula investigations by Benedict Gross and Don Zagier. Explicit small-level Shimura curves computed by John Voight, Aurel Page, and Noam Elkies illustrate rational point phenomena akin to those in Mordell conjecture studies of Gerd Faltings.

Moduli Interpretations and Uniformization

Many Shimura curves admit moduli interpretations as parameter spaces for abelian surfaces with quaternionic multiplication or polarized abelian varieties with extra endomorphism structure, following moduli frameworks of Mumford, Igusa, and Alexei N. Parshin. Uniformization by the upper half-plane connects to complex analytic uniformization theories of Riemann, Bernhard Riemann, and Koebe, while p-adic uniformization results by Christophe Breuil, Pierre Colmez, and Jean-Marc Fontaine relate to rigid-analytic spaces as in Drinfeld uniformization and Cerednik–Drinfeld theory developed by Cherednik and Drinfeld.

Jacobians and Galois Representations

Jacobians of Shimura curves give rise to abelian varieties whose Tate modules realize Galois representations attached to automorphic forms for quaternionic groups, following constructions of Deligne, Carayol, and Taylor–Wiles methods. These Galois representations play roles in modularity lifting theorems of Wiles, Richard Taylor, and Fred Diamond, and link to deformation theory advanced by Barry Mazur and Kazuya Kato. Endomorphism algebras of these Jacobians involve Brandt matrix theory and results of Hecke algebra specialists like Hida and Emerton.

Applications and Open Problems

Shimura curves appear in applications to explicit class field theory as in works of Shimura and Weil, to rational point classification related to Faltings theorem, and to explicit computations in computational number theory by John Cremona and William Stein. Open problems include effective determination of rational points paralleling Birch and Swinnerton-Dyer conjecture questions studied by Birch and Tate, full understanding of special values of L-functions in the spirit of Beilinson conjectures and Bloch–Kato conjecture by Spencer Bloch and Kazuya Kato, and extensions of Langlands reciprocity for non-split quaternionic groups pursued by Robert Langlands and contemporary researchers like Michael Harris and Laurent Clozel.

Category:Algebraic curves