5.1 Complex tori and lattices

Complex tori and lattices form the foundation for understanding elliptic curves from a geometric perspective. These structures bridge the gap between abstract algebra and complex analysis, providing a visual representation of the underlying mathematical concepts.

By studying lattices in the complex plane and their quotients, we gain insights into the periodicity of elliptic functions and the structure of elliptic curves. This connection allows us to explore isomorphisms, endomorphisms, and complex multiplication, deepening our understanding of these fascinating mathematical objects.

Lattices in the complex plane

• Lattices are discrete subgroups of the complex plane that form a regular grid-like structure
• Play a fundamental role in the study of elliptic curves and their properties as complex tori
• Provide a way to construct elliptic functions and study their periodicity

Definition of lattice

• A lattice $\Lambda$ in the complex plane is a discrete subgroup of $\mathbb{C}$ generated by two linearly independent complex numbers $\omega_1$ and $\omega_2$
• Can be written as $\Lambda = \{m\omega_1 + n\omega_2 \mid m,n \in \mathbb{Z}\}$
• The complex numbers $\omega_1$ and $\omega_2$ are called the periods of the lattice
• Example: The Gaussian integers $\mathbb{Z}[i] = \{a + bi \mid a,b \in \mathbb{Z}\}$ form a lattice in $\mathbb{C}$

Basis of a lattice

• A basis of a lattice $\Lambda$ is a pair of complex numbers $(\omega_1, \omega_2)$ that generate the lattice
• The basis is not unique, but any two bases are related by a matrix in $\text{GL}_2(\mathbb{Z})$
• The ratio $\tau = \omega_2/\omega_1$ is called the modular parameter and determines the shape of the lattice up to scaling
• Example: The lattice $\Lambda = \{m + n\sqrt{3}i \mid m,n \in \mathbb{Z}\}$ has a basis $(1, \sqrt{3}i)$

Fundamental parallelogram

• The fundamental parallelogram of a lattice $\Lambda$ with basis $(\omega_1, \omega_2)$ is the set $\{s\omega_1 + t\omega_2 \mid 0 \leq s,t < 1\}$
• Provides a geometric representation of the lattice and its periodicity
• Any point in the complex plane can be uniquely written as a point in the fundamental parallelogram plus a lattice point
• The area of the fundamental parallelogram is given by $|\text{Im}(\omega_1\bar{\omega}_2)|$

Quotient of C by a lattice

• The quotient of the complex plane by a lattice is a key construction in the study of elliptic curves
• Identifies points in the complex plane that differ by a lattice point, resulting in a compact Riemann surface

Torus as quotient space

• The quotient space $\mathbb{C}/\Lambda$ obtained by identifying points in the complex plane that differ by a lattice point is a torus
• The torus inherits a complex structure from the complex plane, making it a compact Riemann surface
• Elliptic curves can be viewed as tori equipped with an algebraic structure

Fundamental domain

• A fundamental domain for the quotient space $\mathbb{C}/\Lambda$ is a subset of $\mathbb{C}$ that contains exactly one representative from each equivalence class
• The fundamental parallelogram is an example of a fundamental domain
• Any meromorphic function on $\mathbb{C}/\Lambda$ can be lifted to a doubly periodic meromorphic function on $\mathbb{C}$

Isomorphic lattices

• Two lattices $\Lambda_1$ and $\Lambda_2$ are isomorphic if there exists a non-zero complex number $\alpha$ such that $\alpha\Lambda_1 = \Lambda_2$
• Isomorphic lattices give rise to isomorphic quotient spaces and elliptic curves
• The modular parameter $\tau$ determines the isomorphism class of a lattice up to scaling

Elliptic functions

• Elliptic functions are meromorphic functions on the complex plane that are doubly periodic with respect to a lattice
• They provide a rich source of examples of meromorphic functions on compact Riemann surfaces
• Elliptic functions are closely related to elliptic curves and play a key role in their study

Doubly periodic functions

• A function $f:\mathbb{C} \to \mathbb{C}$ is doubly periodic with respect to a lattice $\Lambda$ if $f(z+\omega) = f(z)$ for all $z \in \mathbb{C}$ and $\omega \in \Lambda$
• Doubly periodic functions can be viewed as meromorphic functions on the quotient space $\mathbb{C}/\Lambda$
• The periods of a doubly periodic function are the generators of the lattice $\Lambda$

Liouville's theorem

• Liouville's theorem states that any doubly periodic meromorphic function on $\mathbb{C}$ must be constant
• Implies that non-constant elliptic functions must have poles
• The number and order of the poles determine the function up to an additive constant

Weierstrass ℘ function

• The Weierstrass $\wp$ function is a fundamental example of an elliptic function
• Defined as $\wp(z; \Lambda) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \{0\}} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)$
• Has a double pole at each lattice point and satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$
• The coefficients $g_2$ and $g_3$ are invariants of the lattice and determine the isomorphism class of the associated elliptic curve

Elliptic curves as complex tori

• Elliptic curves can be viewed as complex tori equipped with an algebraic structure
• This perspective provides a geometric interpretation of elliptic curves and their properties
• The complex analytic and algebraic definitions of elliptic curves are equivalent

Embedding into projective space

• An elliptic curve $E(\mathbb{C})$ can be embedded into the complex projective plane $\mathbb{P}^2(\mathbb{C})$ using the Weierstrass $\wp$ function and its derivative
• The embedding is given by $z \mapsto [\wp(z) : \wp'(z) : 1]$ for $z \in \mathbb{C} \setminus \Lambda$
• The point at infinity is mapped to $[0 : 1 : 0]$, completing the curve in projective space

Algebraic vs analytic definition

• Algebraically, an elliptic curve is defined as a smooth projective curve of genus one with a specified base point
• Analytically, an elliptic curve is defined as the quotient space $\mathbb{C}/\Lambda$ for some lattice $\Lambda$
• The Weierstrass embedding provides an isomorphism between the algebraic and analytic definitions

Isomorphism classes

• Two elliptic curves $E_1$ and $E_2$ are isomorphic as complex tori if and only if their associated lattices $\Lambda_1$ and $\Lambda_2$ are isomorphic
• The isomorphism class of an elliptic curve is determined by the $j$-invariant, which is a function of the lattice invariants $g_2$ and $g_3$
• Elliptic curves with the same $j$-invariant are isomorphic over $\mathbb{C}$, but may not be isomorphic over other fields

Homomorphisms of complex tori

• Homomorphisms of complex tori are holomorphic maps that preserve the group structure
• They provide a way to study the relationships between different elliptic curves
• Isogenies are a particularly important class of homomorphisms

Complex analytic maps

• A complex analytic map between complex tori $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$ is a holomorphic map that descends from a linear map on the complex plane
• The linear map must satisfy $\alpha\Lambda_1 \subseteq \Lambda_2$ for some complex number $\alpha$
• Complex analytic maps are necessarily group homomorphisms

Isogenies

• An isogeny is a surjective homomorphism between elliptic curves that is also an algebraic morphism
• Isogenies can be viewed as complex analytic maps between the corresponding complex tori
• The degree of an isogeny is the degree of the corresponding algebraic morphism
• Example: Multiplication by $n$ is an isogeny from an elliptic curve to itself for any integer $n$

Isogeny kernels and degrees

• The kernel of an isogeny $\phi: E_1 \to E_2$ is the subgroup $\ker(\phi) = \{P \in E_1 \mid \phi(P) = O\}$, where $O$ is the identity element
• The degree of an isogeny is equal to the order of its kernel
• Isogenies of degree $n$ correspond to index $n$ sublattices of the associated lattice
• Example: The multiplication by $n$ isogeny has kernel isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ and degree $n^2$

Endomorphism rings

• The endomorphism ring of an elliptic curve $E$ is the set of isogenies from $E$ to itself, together with the zero map
• Endomorphism rings provide a way to study the symmetries and arithmetic properties of elliptic curves
• The structure of the endomorphism ring is related to the complex multiplication of the associated lattice

Definition and properties

• The endomorphism ring $\text{End}(E)$ is a ring under pointwise addition and composition of isogenies
• Contains the subring $\mathbb{Z}$ corresponding to multiplication by integers
• Can be viewed as a subring of the ring of algebraic integers in a number field
• The endomorphism ring is either $\mathbb{Z}$ or an order in an imaginary quadratic field

Complex multiplication

• An elliptic curve $E$ has complex multiplication if its endomorphism ring is larger than $\mathbb{Z}$
• Equivalent to the associated lattice $\Lambda$ having additional symmetries beyond scaling
• Elliptic curves with complex multiplication have special arithmetic properties and are important in number theory
• Example: The elliptic curve $y^2 = x^3 - x$ has complex multiplication by the Gaussian integers $\mathbb{Z}[i]$

Orders in imaginary quadratic fields

• If $E$ has complex multiplication, then $\text{End}(E)$ is an order in an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ for some square-free positive integer $d$
• The order is determined by the discriminant of the lattice associated to $E$
• The ring class field of the order is the smallest field over which all elliptic curves with that endomorphism ring are defined
• The Hilbert class field of $K$ is the ring class field of its maximal order and plays a key role in the study of elliptic curves with complex multiplication