Complex multiplication

Complex multiplication
Name	Complex multiplication
Field	Number theory, Algebraic geometry
Introduced	19th century
Notable	Carl Friedrich Gauss, Johann Peter Gustav Lejeune Dirichlet, Henri Poincaré, David Hilbert, André Weil, Goro Shimura, Yutaka Taniyama

Complex multiplication.

Complex multiplication is a theory describing extra endomorphisms of certain complex tori and abelian varieties, especially elliptic curves, that endow them with rich arithmetic structure. It connects classical objects such as imaginary quadratic fields, modular functions, and class field theory with modern concepts in algebraic geometry and automorphic forms. The theory provides explicit class field constructions, guides the study of L-functions, and supplies tools for cryptographic protocols.

Introduction

Complex multiplication arose from work of Carl Friedrich Gauss, Ernst Eduard Kummer, and Johann Peter Gustav Lejeune Dirichlet on binary quadratic forms and class groups, and was shaped by later contributions of Henri Poincaré, David Hilbert, and André Weil. In the elliptic curve setting one studies elliptic curves with endomorphism rings larger than the integers, linking them to imaginary quadratic fields and to the theory of complex multiplication of abelian varieties developed by Goro Shimura and Yutaka Taniyama. The subject intertwines results from the theory of modular functions exemplified by Srinivasa Ramanujan and Bernhard Riemann, and feeds into reciprocity laws envisaged by Emil Artin and formalized in class field theory by Teiji Takagi.

Theory of CM for Elliptic Curves

An elliptic curve over the complex numbers may be expressed as C/Λ for a lattice Λ; when End(C/Λ) strictly contains Z the lattice corresponds to an order in an imaginary quadratic field studied by Carl Friedrich Gauss and Adrien-Marie Legendre. Deuring's theory, building on work by Max Deuring and Ernst Kummer, characterizes such curves by their j-invariants taking algebraic values generating ring class fields of the associated imaginary quadratic field, a connection anticipated by Kronecker and formalized in the Kronecker Jugendtraum influenced by Friedrich Hirzebruch and Heinrich Weber. Complex multiplication implies explicit formulas for Weber and Dedekind eta values featured in tables by Henry Frederick Baker and calculations of singular moduli used by Ramanujan.

CM Fields and Class Field Theory

The principal achievement of complex multiplication is the explicit generation of abelian extensions of imaginary quadratic fields, a realization of ideas of Leopold Kronecker and Ernst Eduard Kummer later systematized by Emil Artin and Teiji Takagi. The theory ties values of modular and elliptic functions at CM points to Hilbert class fields, ring class fields, and ray class fields studied by Helmut Hasse and Richard Dedekind. Shimura and Taniyama extended the reciprocity law to CM fields and constructed reciprocity maps that interact with the Langlands program as developed by Robert Langlands and instantiated in the work of Andrew Wiles on modularity. Class invariants produced by complex multiplication give explicit class field generators compatible with the explicit reciprocity of John Tate and the cohomological methods of Alexander Grothendieck.

Complex Multiplication of Abelian Varieties

Complex multiplication generalizes from elliptic curves to higher-dimensional abelian varieties with CM by CM-fields introduced in the work of André Weil and expanded by Goro Shimura and Yutaka Taniyama. A CM abelian variety has endomorphism algebra isomorphic to a CM-field, a totally imaginary quadratic extension of a totally real field, concepts studied by Ernst Witt and Heinrich Weber. The theory classifies CM types, reflex fields, and CM motives used in the formulation of reciprocity laws by Shimura and the reciprocity conjectures of Pierre Deligne. Complex multiplication abelian varieties produce Hecke characters and algebraic Hecke L-series studied by Harish-Chandra and Atle Selberg.

Explicit Constructions and Examples

Explicit CM constructions date to classical work producing elliptic curves with prescribed complex multiplication via lattices associated to imaginary quadratic orders catalogued by Gauss and tabulated by Karl Friedrich Gauss's successors. Algorithms for computing class invariants and singular moduli utilize modular functions of level structures investigated by Bernhard Riemann and Felix Klein, and computational advances owe to contributions from John von Neumann era numerical analysis and modern implementations inspired by John Tate and Joseph Silverman. Notable examples include elliptic curves with CM by the ring of integers of Q(i) or Q(√−3), connected to special values of modular forms examined by Srinivasa Ramanujan and explicit reciprocity descriptions due to Max Deuring.

Applications in Number Theory and Cryptography

In number theory complex multiplication furnishes explicit class field theory for imaginary quadratic fields, underpins constructions of abelian varieties with prescribed Galois properties used by Andrew Wiles and Richard Taylor in modularity lifting, and informs the study of special values of L-functions addressed by Bernard Gross and Don Zagier. In cryptography CM methods enable generation of pairing-friendly elliptic curves and efficient curve construction algorithms used in protocols developed by Victor Miller and Dan Boneh; implementations draw on parameter-selection strategies by Niels Henrik Abel-inspired theory and computational techniques refined by Aubrey de Grey-era software. Complex multiplication also appears in explicit point-counting algorithms by Robert Schoof and refinements by Noam Elkies and Andrew Ogg.

Category:Algebraic number theory