5.5 Elliptic curve uniformization

Elliptic curve uniformization connects complex analysis, algebraic geometry, and number theory. It reveals how elliptic curves over complex numbers can be viewed as quotients of the complex plane by lattices, providing deep insights into their structure and properties.

This perspective allows us to understand elliptic curves through the lens of complex tori and Weierstrass functions. It forms the foundation for studying isomorphisms, complex multiplication, and connections to modular forms and elliptic functions.

Elliptic curves over complex numbers

• Elliptic curves over the complex numbers provide a rich and beautiful interplay between complex analysis, algebraic geometry, and number theory
• The study of elliptic curves over C allows for a deep understanding of their geometric and analytic properties, which can then be applied to other fields such as cryptography and mathematical physics

Lattices in the complex plane

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• A lattice in the complex plane is a discrete subgroup of C of the form $L = \{m\omega_1 + n\omega_2 : m,n \in \mathbb{Z}\}$, where $\omega_1, \omega_2 \in \mathbb{C}$ are linearly independent over $\mathbb{R}$
• The generators $\omega_1$ and $\omega_2$ are called the periods of the lattice
• The fundamental parallelogram of a lattice is the set $\{a\omega_1 + b\omega_2 : 0 \leq a,b < 1\}$, which tiles the complex plane under translations by lattice elements
• The shape of the fundamental parallelogram determines the geometry of the lattice (square, rectangular, rhombic)

Quotient of C by lattice

• The quotient of the complex plane C by a lattice L, denoted $\mathbb{C}/L$, is the set of equivalence classes of complex numbers under the equivalence relation $z_1 \sim z_2$ if and only if $z_1 - z_2 \in L$
• The quotient $\mathbb{C}/L$ inherits a natural complex analytic structure from C, making it a compact Riemann surface
• The quotient $\mathbb{C}/L$ is topologically equivalent to a torus, obtained by identifying opposite sides of the fundamental parallelogram

Elliptic curves as complex tori

• An elliptic curve over C can be realized as a complex torus $\mathbb{C}/L$ for some lattice L
• The group law on the elliptic curve corresponds to the natural group structure on the torus, with the point at infinity serving as the identity element
• The Weierstrass $\wp$-function associated to the lattice L provides a parametrization of the elliptic curve, giving an explicit isomorphism between $\mathbb{C}/L$ and the curve in Weierstrass form

Uniformization theorem

• The uniformization theorem is a fundamental result in the theory of Riemann surfaces, providing a classification of simply connected Riemann surfaces
• In the context of elliptic curves, the uniformization theorem states that every elliptic curve over C is analytically isomorphic to a complex torus $\mathbb{C}/L$ for some lattice L

Statement of the theorem

• Let E be an elliptic curve over C. Then there exists a lattice L in C such that E is analytically isomorphic to the complex torus $\mathbb{C}/L$
• The isomorphism is given by the Weierstrass $\wp$-function associated to the lattice L
• The lattice L is uniquely determined by E up to homothety (scaling by a complex number)

Existence of Weierstrass parametrization

• The Weierstrass $\wp$-function associated to a lattice L is defined as $\wp(z) = \frac{1}{z^2} + \sum_{\omega \in L \setminus \{0\}} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)$
• The $\wp$-function is an even, meromorphic function on C with double poles at the lattice points and satisfies the differential equation $(\wp'(z))^2 = 4\wp(z)^3 - g_2\wp(z) - g_3$, where $g_2$ and $g_3$ are constants depending on the lattice L
• The map $\mathbb{C}/L \to E$ given by $z \mapsto (\wp(z), \wp'(z))$ is an analytic isomorphism between the complex torus and the elliptic curve E in Weierstrass form $y^2 = 4x^3 - g_2x - g_3$

Uniqueness up to isomorphism

• If two elliptic curves $E_1$ and $E_2$ over C are analytically isomorphic, then their corresponding lattices $L_1$ and $L_2$ are homothetic, meaning there exists a complex number $\alpha$ such that $L_2 = \alpha L_1$
• Conversely, if two lattices $L_1$ and $L_2$ are homothetic, then the corresponding elliptic curves $\mathbb{C}/L_1$ and $\mathbb{C}/L_2$ are analytically isomorphic
• This uniqueness property allows for the classification of elliptic curves over C up to isomorphism by their associated lattices

Weierstrass ℘ function

• The Weierstrass $\wp$-function is a central object in the study of elliptic curves over the complex numbers, providing a bridge between the analytic and algebraic aspects of the theory
• The $\wp$-function is a meromorphic function on C that is doubly periodic with respect to a lattice L

Definition and properties

• The Weierstrass $\wp$-function associated to a lattice L is defined as $\wp(z) = \frac{1}{z^2} + \sum_{\omega \in L \setminus \{0\}} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)$
• $\wp(z)$ is an even function, satisfying $\wp(-z) = \wp(z)$
• $\wp(z)$ is periodic with respect to the lattice L, meaning $\wp(z+\omega) = \wp(z)$ for all $\omega \in L$
• $\wp(z)$ has double poles at the lattice points and no other poles

Relation to elliptic curves

• The Weierstrass $\wp$-function parametrizes an elliptic curve in Weierstrass form $y^2 = 4x^3 - g_2x - g_3$, where $g_2$ and $g_3$ are constants depending on the lattice L
• The map $\phi: \mathbb{C}/L \to E$ given by $\phi(z) = (\wp(z), \wp'(z))$ is an analytic isomorphism between the complex torus $\mathbb{C}/L$ and the elliptic curve E
• The group law on the elliptic curve E corresponds to the natural group structure on the torus $\mathbb{C}/L$ under the isomorphism $\phi$

Differential equation satisfied by ℘

• The Weierstrass $\wp$-function satisfies the differential equation $(\wp'(z))^2 = 4\wp(z)^3 - g_2\wp(z) - g_3$, where $g_2$ and $g_3$ are constants depending on the lattice L
• This differential equation characterizes the $\wp$-function and plays a crucial role in the study of elliptic curves and their associated functions
• The constants $g_2$ and $g_3$ are called the invariants of the Weierstrass equation and determine the isomorphism class of the elliptic curve

Periods and quasi-periods

• The periods and quasi-periods of an elliptic curve are fundamental quantities associated with the lattice uniformization of the curve
• They encode important information about the geometry and arithmetic of the elliptic curve

Periods of elliptic functions

• The periods of an elliptic function $f$ with respect to a lattice L are the complex numbers $\omega \in L$ such that $f(z+\omega) = f(z)$ for all $z \in \mathbb{C}$
• For the Weierstrass $\wp$-function, the periods are precisely the elements of the lattice L
• The periods form a discrete subgroup of C and generate the lattice L

Quasi-periods and the period lattice

• The quasi-periods of the Weierstrass $\wp$-function are the values of the integral $\eta_i = \int_0^{\omega_i} \wp(z) dz$, where $\omega_1$ and $\omega_2$ are the fundamental periods of the lattice L
• The quasi-periods satisfy the relation $\eta_1\omega_2 - \eta_2\omega_1 = \pm 2\pi i$
• The period lattice of an elliptic curve is the lattice generated by the periods $\omega_1, \omega_2$ and the quasi-periods $\eta_1, \eta_2$

Legendre relation for periods

• The periods $\omega_1, \omega_2$ and quasi-periods $\eta_1, \eta_2$ of an elliptic curve satisfy the Legendre relation $\eta_1\omega_2 - \eta_2\omega_1 = 2\pi i$
• This relation is a consequence of the residue theorem applied to the Weierstrass $\wp$-function
• The Legendre relation plays a crucial role in the theory of elliptic integrals and the study of the moduli space of elliptic curves

Isomorphisms of elliptic curves

• Isomorphisms of elliptic curves over the complex numbers are an important tool for understanding the geometry and arithmetic of these curves
• They allow for the classification of elliptic curves up to isomorphism and the study of their symmetries

Isomorphisms over C

• Two elliptic curves $E_1$ and $E_2$ over C are isomorphic if there exists a complex analytic isomorphism $\phi: E_1 \to E_2$ that preserves the group structure
• Isomorphic elliptic curves have the same j-invariant, a complex number that characterizes the isomorphism class of the curve
• Conversely, two elliptic curves with the same j-invariant are isomorphic over C

Isomorphisms preserving the group law

• An isomorphism $\phi: E_1 \to E_2$ between elliptic curves is said to preserve the group law if $\phi(P+Q) = \phi(P) + \phi(Q)$ for all points $P,Q \in E_1$
• Isomorphisms preserving the group law are precisely the complex analytic isomorphisms between the curves
• The set of isomorphisms preserving the group law forms a group under composition, called the automorphism group of the elliptic curve

Automorphisms and endomorphisms

• An automorphism of an elliptic curve E is an isomorphism $\phi: E \to E$ from the curve to itself that preserves the group law
• The set of automorphisms of E forms a finite group, denoted $\text{Aut}(E)$, which is either cyclic of order 2, 4, or 6, or the Klein four-group
• An endomorphism of an elliptic curve E is a complex analytic map $\phi: E \to E$ that preserves the group law, but may not be an isomorphism
• The set of endomorphisms of E forms a ring under pointwise addition and composition, called the endomorphism ring of E

Complex multiplication

• Complex multiplication is a special property that certain elliptic curves over the complex numbers possess, which has far-reaching consequences in number theory and cryptography
• Elliptic curves with complex multiplication have additional symmetries and arithmetic properties that make them particularly interesting to study

Elliptic curves with complex multiplication

• An elliptic curve E over C is said to have complex multiplication (CM) if its endomorphism ring $\text{End}(E)$ is strictly larger than $\mathbb{Z}$
• Equivalently, E has CM if there exists an imaginary quadratic field K such that $K \subseteq \text{End}(E) \otimes_\mathbb{Z} \mathbb{Q}$
• Elliptic curves with CM are rare; they form a countable subset of the moduli space of elliptic curves

CM by quadratic imaginary fields

• If an elliptic curve E has CM by an imaginary quadratic field K, then K is isomorphic to a subfield of the endomorphism algebra $\text{End}(E) \otimes_\mathbb{Z} \mathbb{Q}$
• The conductor of the order $\mathcal{O} = \text{End}(E)$ in K is an important invariant of the curve E, measuring the complexity of its CM structure
• Elliptic curves with CM by the same imaginary quadratic field and conductor are isogenous, meaning there exists a non-constant rational map between them that preserves the group law

Hilbert class field and ray class fields

• The Hilbert class field of an imaginary quadratic field K is the maximal unramified abelian extension of K
• Elliptic curves with CM by K are closely related to the Hilbert class field of K; their j-invariants generate the Hilbert class field over K
• Ray class fields are generalizations of the Hilbert class field that take into account the conductor of the CM order
• Elliptic curves with CM by K and a given conductor are related to the corresponding ray class field of K

Modular functions and modular forms

• Modular functions and modular forms are central objects in the theory of elliptic curves and number theory
• They provide a powerful tool for understanding the arithmetic and geometric properties of elliptic curves and their moduli spaces

Modular functions and modular curves

• A modular function is a meromorphic function on the upper half-plane $\mathcal{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$ that is invariant under the action of a congruence subgroup $\Gamma$ of $\text{SL}_2(\mathbb{Z})$
• The quotient space $\Gamma \backslash \mathcal{H}^*$, where $\mathcal{H}^* = \mathcal{H} \cup \mathbb{Q} \cup \{\infty\}$, is a compact Riemann surface called a modular curve
• Modular functions can be thought of as meromorphic functions on modular curves
• The j-invariant of an elliptic curve is an example of a modular function on the full modular group $\text{SL}_2(\mathbb{Z})$

Eisenstein series and cusp forms

• Eisenstein series are special modular forms that are eigenfunctions of the Laplace operator on the upper half-plane
• They play a crucial role in the theory of modular forms and the study of the arithmetic of elliptic curves
• Cusp forms are modular forms that vanish at the cusps (the points at infinity) of a modular curve
• The space of cusp forms of a given weight and level is a finite-dimensional vector space with rich arithmetic properties

Hecke operators and Hecke eigenforms

• Hecke operators are linear operators acting on the space of modular forms of a given weight and level
• They are defined using the action of certain double cosets of the congruence subgroup on the upper half-plane
• Hecke eigenforms are modular forms that are simultaneous eigenfunctions of all Hecke operators
• The Fourier coefficients of Hecke eigenforms encode important arithmetic information, such as the number of points on elliptic curves over finite fields

Elliptic functions and elliptic integrals

• Elliptic functions and elliptic integrals are closely related to the theory of elliptic curves and provide a rich source of examples and applications
• They have a long history in mathematics and have been studied extensively for their analytic and geometric properties

Elliptic integrals of the first kind

• An elliptic integral of the first kind is an integral of the form $\int \frac{dx}{\sqrt{P(x)}}$, where $P(x)$ is a cubic or quartic polynomial with distinct roots
• These integrals arise naturally in the study of the arc length of ellipses and the period lattice of elliptic curves
• The complete elliptic integral of the first kind, denoted $K(m)$, is a special case where the limits of integration are 0 and 1, and $P(x) = (1-x^2)(1-mx^2)$

Elliptic integrals of the second kind

• An elliptic integral of the second kind is an integral of the form $\int \sqrt{P(x)} dx$, where $P(x)$ is a cubic or quartic polynomial with distinct roots
• These integrals arise in the study of the area enclosed by ellipses and the quasi-periods of ell