2.6 Complex multiplication

Complex multiplication intertwines algebraic number theory and elliptic curves, providing powerful tools for studying their arithmetic properties. It plays a crucial role in understanding elliptic curves and their applications in cryptography.

The theory extends from basic properties of complex number multiplication to endomorphism rings of elliptic curves. For CM elliptic curves, these rings are isomorphic to orders in imaginary quadratic fields, leading to rich connections with class field theory and modular forms.

Fundamentals of complex multiplication

• Complex multiplication forms a cornerstone of arithmetic geometry intertwining algebraic number theory and elliptic curves
• Provides powerful tools for studying algebraic structures and number-theoretic properties of elliptic curves
• Plays a crucial role in understanding the arithmetic of elliptic curves and their applications in cryptography

Definition and basic properties

Top images from around the web for Definition and basic properties

• 

  Euler's formula - Wikipedia View original

  Is this image relevant?

• 

  Polar Form of Complex Numbers – Algebra and Trigonometry OpenStax View original

  Is this image relevant?

• 

  Exercise : Complex Number | Math2Ever - Blog Politeknik Malaysia View original

  Is this image relevant?

• 

  Euler's formula - Wikipedia View original

  Is this image relevant?

• 

  Polar Form of Complex Numbers – Algebra and Trigonometry OpenStax View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and basic properties

• 

  Euler's formula - Wikipedia View original

  Is this image relevant?

• 

  Polar Form of Complex Numbers – Algebra and Trigonometry OpenStax View original

  Is this image relevant?

• 

  Exercise : Complex Number | Math2Ever - Blog Politeknik Malaysia View original

  Is this image relevant?

• 

  Euler's formula - Wikipedia View original

  Is this image relevant?

• 

  Polar Form of Complex Numbers – Algebra and Trigonometry OpenStax View original

  Is this image relevant?

1 of 3

• Multiplication of complex numbers $z_1 = a + bi$ and $z_2 = c + di$ defined as $(ac - bd) + (ad + bc)i$
• Commutative property holds for complex multiplication: $z_1 * z_2 = z_2 * z_1$
• Distributive property applies: $z_1 * (z_2 + z_3) = z_1 * z_2 + z_1 * z_3$
• Polar form representation: $z = r(\cos\theta + i\sin\theta)$ simplifies multiplication to $r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$

Complex multiplication in elliptic curves

• Refers to endomorphisms of an elliptic curve $E$ over a number field $K$
• Endomorphism ring $End(E)$ contains more than just the multiplication-by-n maps
• CM elliptic curves have an endomorphism ring isomorphic to an order in an imaginary quadratic field
• Characterized by the existence of a non-trivial endomorphism $\alpha: E \rightarrow E$ not induced by multiplication by an integer

Endomorphism rings

• Ring of endomorphisms of an elliptic curve $E$ denoted as $End(E)$
• For general elliptic curves, $End(E) \cong \mathbb{Z}$
• CM elliptic curves have larger endomorphism rings, isomorphic to an order in an imaginary quadratic field
• Structure of $End(E)$ determines important arithmetic properties of the elliptic curve
• Classification of possible endomorphism rings leads to the theory of CM types

CM fields and class field theory

• CM fields provide the algebraic framework for studying complex multiplication on elliptic curves
• Class field theory offers powerful tools for understanding the arithmetic of CM fields
• Connects complex multiplication to broader areas of algebraic number theory and arithmetic geometry

Quadratic imaginary fields

• Number fields of the form $\mathbb{Q}(\sqrt{-d})$ where $d$ is a positive square-free integer
• Fundamental discriminant $D$ determines the field: $D = -d$ if $d \equiv 3 \pmod{4}$, or $D = -4d$ otherwise
• Ring of integers $\mathcal{O}_K = \mathbb{Z}[\frac{D + \sqrt{D}}{2}]$ for $D \equiv 1 \pmod{4}$, or $\mathbb{Z}[\sqrt{D}]$ otherwise
• Class number $h(D)$ measures the failure of unique factorization in $\mathcal{O}_K$
• Crucial for CM theory as endomorphism rings of CM elliptic curves are orders in these fields

Hilbert class fields

• Maximal unramified abelian extension $H$ of a number field $K$
• For imaginary quadratic fields, $H$ corresponds to the field of moduli of CM elliptic curves
• Galois group $Gal(H/K)$ isomorphic to the ideal class group of $K$
• Degree $[H:K]$ equals the class number $h(D)$ of the base field
• Generation of $H$ using special values of modular functions (singular moduli)

Ring class fields

• Generalizes Hilbert class fields for non-maximal orders in a number field
• For an order $\mathcal{O}$ in $K$, the ring class field $H_\mathcal{O}$ is an abelian extension of $K$
• $Gal(H_\mathcal{O}/K)$ isomorphic to the ideal class group of $\mathcal{O}$
• Corresponds to moduli of elliptic curves with CM by the order $\mathcal{O}$
• Crucial for understanding the full spectrum of CM elliptic curves over a given field

Complex multiplication and modular forms

• Modular forms provide a powerful framework for studying CM elliptic curves
• CM points on modular curves correspond to elliptic curves with complex multiplication
• Connects complex multiplication to the rich theory of automorphic forms and L-functions

CM points on modular curves

• Special points on modular curves $X_0(N)$ corresponding to elliptic curves with CM
• Coordinates of CM points given by special values of the j-invariant (singular moduli)
• CM points form Galois orbits under the action of ideal class groups
• Crucial for understanding the arithmetic of CM elliptic curves and their reductions
• Heegner points arise as CM points on modular curves of higher level

Singular moduli

• Special values of the j-function at CM points
• Generate class fields of imaginary quadratic fields: $H = K(j(\tau))$ for $\tau$ a CM point
• Satisfy polynomial equations with coefficients in the base field $K$
• Class polynomial $H_D(X)$ has roots precisely the singular moduli for discriminant $D$
• Applications in explicit class field theory and construction of CM elliptic curves

Heegner points

• Special points on modular curves $X_0(N)$ associated to quadratic imaginary fields
• Arise from optimal embeddings of orders into quaternion algebras
• Play a crucial role in the Birch and Swinnerton-Dyer conjecture for elliptic curves
• Used to construct rational points on elliptic curves (Heegner point construction)
• Connection to L-functions of elliptic curves through the Gross-Zagier formula

Arithmetic applications

• Complex multiplication provides powerful tools for solving various arithmetic problems
• Applications span from classical number theory to modern cryptography
• Demonstrates the deep connections between abstract theory and practical computations

Class number formulas

• Relate class numbers of imaginary quadratic fields to special values of L-functions
• Dirichlet's class number formula: $h(D) = \frac{\sqrt{|D|}}{2\pi} L(1, \chi_D)$ where $\chi_D$ is the quadratic character
• Kronecker's limit formula connects class numbers to logarithms of singular moduli
• Genus theory provides lower bounds and congruence conditions for class numbers
• Applications in determining imaginary quadratic fields with class number one (Baker's theorem)

Generation of class fields

• Complex multiplication provides explicit methods for generating class fields
• Hilbert class field $H$ of $K = \mathbb{Q}(\sqrt{-d})$ generated by $j(\tau)$ where $\tau = \frac{-b + \sqrt{-d}}{2a}$
• Ring class fields generated by singular moduli corresponding to non-maximal orders
• Weber functions provide more efficient generators in some cases
• Explicit construction of abelian extensions crucial for computational number theory

Elliptic curve cryptography

• CM method for generating cryptographically secure elliptic curves
• Allows precise control over the group order for efficient and secure implementations
• Construction of curves with prescribed embedding degree for pairing-based cryptography
• Efficient point counting on CM curves using the CM method
• Trade-off between efficiency of curve generation and potential security vulnerabilities

CM for higher-dimensional abelian varieties

• Generalizes complex multiplication theory from elliptic curves to higher-dimensional abelian varieties
• Provides a rich framework for studying arithmetic properties of abelian varieties
• Connects to broader areas of algebraic geometry and number theory through Shimura varieties

Abelian surfaces with CM

• Two-dimensional analogue of CM elliptic curves
• Endomorphism algebra is a quartic CM field or a product of two imaginary quadratic fields
• Classification of CM types for abelian surfaces: simple and non-simple types
• Polarizations and the Rosati involution play crucial roles in the theory
• Applications in constructing cryptographically suitable genus 2 curves

Shimura varieties

• Moduli spaces of abelian varieties with additional structures (polarizations, level structures)
• Generalize modular curves to higher dimensions
• CM points on Shimura varieties correspond to abelian varieties with complex multiplication
• Special subvarieties and their Galois orbits crucial for the André-Oort conjecture
• Connection to the theory of automorphic forms and representations

Period matrices

• Encode complex analytic information of abelian varieties
• For CM abelian varieties, entries of period matrices related to special values of CM type
• Mumford-Tate group determined by the CM type reflects in the structure of the period matrix
• Torelli's theorem establishes bijection between principally polarized abelian varieties and period matrices
• Computational aspects: numerical approximation of period matrices for CM abelian varieties

Computational aspects

• Implementing complex multiplication theory requires sophisticated algorithms and software
• Balances theoretical understanding with practical computational challenges
• Essential for applications in cryptography and computational number theory

Algorithms for CM

• Class polynomial computation: complex analytic, p-adic, Chinese Remainder Theorem methods
• Explicit CM construction of elliptic curves over finite fields
• Point counting on CM elliptic curves using the CM method
• Generation of CM abelian varieties in higher dimensions
• Computation of Hilbert class polynomials and ring class polynomials

Implementation challenges

• Precision requirements for floating-point computations in complex analytic methods
• Efficient arithmetic in number fields and function fields
• Memory management for large degree class polynomials
• Parallelization strategies for computationally intensive tasks
• Numerical stability issues in period matrix computations for higher-dimensional cases

Software packages for CM

• PARI/GP provides extensive functionality for CM computations and class field theory
• Sage integrates various CM algorithms and interfaces with specialized libraries
• Magma offers powerful tools for CM computations, especially in higher dimensions
• CM software by David Kohel for specialized CM elliptic curve constructions
• RECIP package for relative class field computations and CM applications

Connections to other areas

• Complex multiplication interacts with various branches of number theory and algebraic geometry
• Provides insights into deeper structures and conjectures in arithmetic geometry
• Serves as a bridge between classical theory and modern developments in the field

CM vs ordinary reduction

• CM elliptic curves have potential good reduction at all primes
• Ordinary reduction occurs at primes splitting in the CM field
• Supersingular reduction at primes inert or ramified in the CM field
• Density of ordinary primes determined by the CM field (Deuring's criterion)
• Applications in studying reductions of higher-dimensional abelian varieties

CM and Galois representations

• Galois representations attached to CM elliptic curves have a particularly simple structure
• Decomposition of the ℓ-adic Tate module reflects the CM structure
• Induces abelian Galois representations with values in the CM field
• Connects to Serre's open image theorem and its generalizations
• Applications in studying Galois representations of higher-dimensional CM abelian varieties

CM and L-functions

• L-functions of CM elliptic curves factor as a product of Hecke L-functions
• Allows explicit computation of special L-values in some cases
• Connects to the Birch and Swinnerton-Dyer conjecture through Heegner points
• Gross-Zagier formula relates heights of Heegner points to derivatives of L-functions
• Generalizations to higher-dimensional CM abelian varieties and their L-functions

Advanced topics

• Explores cutting-edge research areas involving complex multiplication
• Connects CM theory to modern developments in arithmetic geometry and number theory
• Provides directions for future research and applications of CM theory

CM and p-adic Hodge theory

• Describes p-adic Galois representations attached to CM abelian varieties
• Crystalline representations arise from good reduction at p
• Explicit description of Hodge-Tate decomposition for CM varieties
• Connections to p-adic periods and p-adic L-functions
• Applications in studying p-adic properties of special values of L-functions

CM and motives

• CM abelian varieties provide concrete examples of motives
• Realizations (Betti, de Rham, ℓ-adic) have particularly simple structures for CM motives
• Mumford-Tate conjecture known for CM motives
• Connections to Deligne's period conjecture and transcendence theory
• Generalizations to mixed motives and their periods

Recent developments in CM theory

• Higher-dimensional generalizations of CM theory (Shimura varieties)
• Connections to the André-Oort conjecture and unlikely intersections
• CM cycles and their applications in arithmetic intersection theory
• Developments in explicit class field theory using CM methods
• Applications of CM theory in quantum computing and post-quantum cryptography