7.1 Complex tori

Complex tori bridge complex analysis and algebraic geometry, providing concrete examples of abelian varieties over complex numbers. They illustrate connections between elliptic curves, modular forms, and number theory, serving as fundamental objects in arithmetic geometry.

Defined as quotients of the complex plane by lattices, complex tori have rich topological and algebraic structures. They represent all genus 1 compact Riemann surfaces and, when algebraic, correspond to elliptic curves over C, enabling the study of their arithmetic properties.

Definition of complex tori

• Complex tori serve as fundamental objects in arithmetic geometry bridging complex analysis and algebraic geometry
• Provide concrete examples of abelian varieties over the complex numbers crucial for studying arithmetic properties
• Illustrate deep connections between elliptic curves, modular forms, and number theory

Lattices in complex plane

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• Discrete subgroups of $\mathbb{C}$ generated by two $\mathbb{R}$-linearly independent complex numbers $\omega_1$ and $\omega_2$
• Expressed as $\Lambda = \{m\omega_1 + n\omega_2 : m,n \in \mathbb{Z}\}$
• Fundamental parallelogram represents the basic building block of the lattice
• Different choices of basis vectors yield equivalent lattices (related by $SL_2(\mathbb{Z})$ transformations)

Quotient space construction

• Complex torus defined as the quotient space $\mathbb{C}/\Lambda$
• Identifies points in $\mathbb{C}$ that differ by elements of the lattice $\Lambda$
• Resulting space inherits complex structure from $\mathbb{C}$
• Can be visualized as "wrapping" the complex plane around a donut shape

Topological structure

• Homeomorphic to the product of two circles $S^1 \times S^1$
• Compact and connected 2-dimensional real manifold (genus 1 surface)
• Admits a flat metric inherited from the complex plane
• Fundamental group isomorphic to $\mathbb{Z}^2$, reflecting the two "holes" in the torus

Properties of complex tori

• Serve as prototypical examples of compact complex manifolds in dimension 1
• Illustrate interplay between complex analysis, topology, and algebra in arithmetic geometry
• Provide concrete realizations of abstract concepts like sheaf cohomology and Hodge theory

Compact Riemann surfaces

• Complex tori represent all compact Riemann surfaces of genus 1
• Holomorphic functions on complex tori are constant (Liouville's theorem)
• Meromorphic functions form a field of transcendence degree 1 over $\mathbb{C}$
• Possess a unique (up to isomorphism) complex analytic group structure

Algebraic vs analytic tori

• Every complex torus is an analytic variety by construction
• Not all complex tori are algebraic varieties (projective algebraic curves)
• Algebraic complex tori precisely correspond to elliptic curves over $\mathbb{C}$
• Characterization via existence of sufficiently many meromorphic functions (Riemann's theorem)

Group structure

• Complex tori inherit natural abelian group structure from $\mathbb{C}$
• Addition law defined by $(z_1 + \Lambda) + (z_2 + \Lambda) = (z_1 + z_2) + \Lambda$
• Identity element represented by $0 + \Lambda$
• Group operation holomorphic, making complex tori complex Lie groups

Complex tori as elliptic curves

• Establish fundamental connection between complex analysis and algebraic geometry
• Enable study of arithmetic properties of elliptic curves using complex analytic methods
• Provide geometric intuition for abstract concepts in the theory of elliptic curves

Weierstrass ℘-function

• Doubly periodic meromorphic function associated to a lattice $\Lambda$
• Defined as $\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \{0\}} (\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2})$
• Satisfies differential equation $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$ where $g_2$ and $g_3$ are lattice invariants
• Generates all meromorphic functions on the complex torus $\mathbb{C}/\Lambda$

Uniformization theorem

• States every elliptic curve over $\mathbb{C}$ is isomorphic to a complex torus
• Provides explicit isomorphism $\mathbb{C}/\Lambda \to E(\mathbb{C})$ via $z \mapsto [\wp(z):\wp'(z):1]$
• Allows translation between analytic and algebraic descriptions of elliptic curves
• Fundamental tool for studying arithmetic properties of elliptic curves over number fields

j-invariant

• Complex analytic invariant that uniquely determines isomorphism class of complex tori
• Defined as $j(\Lambda) = 1728 \frac{g_2^3}{g_2^3 - 27g_3^2}$ where $g_2$ and $g_3$ are Eisenstein series
• Takes values in $\mathbb{C}$ and is invariant under $SL_2(\mathbb{Z})$ transformations of the lattice
• Algebraic over $\mathbb{Q}$ if and only if the corresponding elliptic curve has complex multiplication

Moduli space of complex tori

• Parametrizes isomorphism classes of complex tori
• Crucial for understanding families of elliptic curves and their degenerations
• Provides geometric setting for studying modular forms and arithmetic of elliptic curves

Fundamental domain

• Region in the upper half-plane $\mathbb{H}$ representing distinct isomorphism classes of complex tori
• Standard choice $\{z \in \mathbb{H} : |z| \geq 1, -\frac{1}{2} < \Re(z) \leq \frac{1}{2}\}$
• Quotient of $\mathbb{H}$ by action of $SL_2(\mathbb{Z})$ yields moduli space $\mathbb{H}/SL_2(\mathbb{Z})$
• Compactified by adding cusp at infinity, resulting in modular curve $X(1)$

Modular forms

• Holomorphic functions on $\mathbb{H}$ satisfying transformation law under $SL_2(\mathbb{Z})$
• Eisenstein series $G_k(\tau)$ provide important examples of modular forms
• Spaces of modular forms finite-dimensional and graded by weight
• Crucial for studying arithmetic properties of elliptic curves (modularity theorem)

Elliptic modular surface

• Universal family of elliptic curves over the modular curve $X(1)$
• Fiber over each point represents the corresponding isomorphism class of complex tori
• Singular fibers correspond to degenerate elliptic curves (nodal or cuspidal)
• Provides geometric realization of the theory of elliptic curves over varying base fields

Isogenies between complex tori

• Represent surjective holomorphic homomorphisms between complex tori
• Crucial for studying relationships between different elliptic curves
• Play fundamental role in arithmetic of elliptic curves and abelian varieties

Definition and properties

• Holomorphic map $f: \mathbb{C}/\Lambda_1 \to \mathbb{C}/\Lambda_2$ induced by complex linear map $\alpha: \mathbb{C} \to \mathbb{C}$
• Satisfies $\alpha(\Lambda_1) \subseteq \Lambda_2$ and has finite kernel
• Degree of isogeny defined as $[\Lambda_2 : \alpha(\Lambda_1)]$, always finite and positive
• Preserve group structure and complex analytic structure of tori

Dual isogenies

• For every isogeny $f: T_1 \to T_2$, there exists a dual isogeny $\hat{f}: T_2 \to T_1$
• Satisfy relations $\hat{f} \circ f = [n]_{T_1}$ and $f \circ \hat{f} = [n]_{T_2}$ where $n = \deg(f)$
• Provide duality theory for isogenies analogous to dual abelian varieties
• Essential for studying Weil pairings and Tate modules of elliptic curves

Endomorphism ring

• Ring of all isogenies from a complex torus to itself, denoted $\text{End}(T)$
• Always contains $\mathbb{Z}$ (multiplication-by-n maps)
• Larger endomorphism rings indicate special arithmetic properties (complex multiplication)
• Isomorphic to orders in imaginary quadratic fields for CM elliptic curves

Complex multiplication

• Study of complex tori with larger-than-usual endomorphism rings
• Connects class field theory, elliptic curves, and algebraic number theory
• Provides explicit class field theory for imaginary quadratic fields

CM fields

• Imaginary quadratic extensions of $\mathbb{Q}$ (quadratic fields with negative discriminant)
• Examples include $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt{-3})$, $\mathbb{Q}(\sqrt{-7})$
• Endomorphism ring of CM elliptic curve isomorphic to an order in a CM field
• Determine arithmetic properties of corresponding elliptic curves

Class field theory connection

• CM elliptic curves generate abelian extensions of their CM fields
• j-invariants of CM elliptic curves generate Hilbert class fields of CM fields
• Provides explicit construction of class fields using analytic methods
• Realizes Kronecker's vision of using elliptic functions to solve class field theory

Kronecker's Jugendtraum

• Vision of using elliptic functions to solve problems in algebraic number theory
• Sought to generalize theory of complex multiplication to higher-dimensional abelian varieties
• Led to development of class field theory and theory of complex multiplication
• Continues to inspire research in arithmetic geometry and number theory

Periods and period lattices

• Encode essential arithmetic and geometric information about complex tori
• Provide bridge between complex analysis and algebraic geometry
• Crucial for studying transcendence properties and special values of L-functions

Period integrals

• Integrals of holomorphic differentials along closed paths on the complex torus
• Generate the period lattice $\Lambda = \{\int_\gamma \omega : \gamma \in H_1(T, \mathbb{Z})\}$
• Depend on choice of basis for $H_1(T, \mathbb{Z})$ and holomorphic differential $\omega$
• Satisfy relations encoded in the period matrix of the complex torus

Transcendence properties

• Periods of complex tori generally transcendental (non-algebraic) complex numbers
• Algebraic relations between periods governed by Grothendieck's period conjecture
• Special values (ratios of periods) can have arithmetic significance (CM case)
• Study of transcendence properties connects to Schanuel's conjecture and other deep problems

Relation to L-functions

• Periods appear in special values of L-functions associated to complex tori
• Critical values of L-functions often expressible in terms of periods (up to algebraic factors)
• Birch and Swinnerton-Dyer conjecture relates L-function values to arithmetic of elliptic curves
• Periods crucial for formulating and studying conjectures on special values of L-functions

Applications in arithmetic geometry

• Complex tori provide concrete examples for studying deep phenomena in arithmetic geometry
• Serve as testing ground for conjectures and techniques applicable to more general settings
• Illustrate interplay between complex analysis, algebraic geometry, and number theory

Mordell-Weil theorem

• States that the group of rational points $E(\mathbb{Q})$ on an elliptic curve $E$ is finitely generated
• Proof uses height functions and descent theory on complex tori
• Generalizes to abelian varieties over number fields
• Fundamental result connecting arithmetic and geometry of elliptic curves

Height functions

• Real-valued functions measuring arithmetic complexity of points on elliptic curves
• Néron-Tate height derived from complex analytic Néron-Tate pairing
• Satisfy crucial properties (quadraticity, positivity) used in proof of Mordell-Weil theorem
• Provide tool for studying distribution of rational points on elliptic curves

Néron-Tate pairing

• Bilinear pairing on $E(\overline{\mathbb{Q}})$ taking values in $\mathbb{R}$
• Defined using complex analytic methods (Green's functions on Riemann surfaces)
• Satisfies important properties (positive definiteness on $E(\mathbb{Q}) \otimes \mathbb{R}$)
• Crucial for studying arithmetic properties of elliptic curves (ranks, regulators)

Torsion points on complex tori

• Finite-order points in the group structure of complex tori
• Provide deep connections between arithmetic, geometry, and Galois theory
• Essential for studying rational points and Galois representations attached to elliptic curves

Structure theorem

• Torsion subgroup of complex torus isomorphic to $(\mathbb{Q}/\mathbb{Z})^2$
• n-torsion points form group isomorphic to $(\mathbb{Z}/n\mathbb{Z})^2$
• Explicit description: $E[n] = \{\frac{a\omega_1 + b\omega_2}{n} : 0 \leq a,b < n\}$ for lattice basis $\{\omega_1, \omega_2\}$
• Crucial for studying isogenies and Galois representations attached to elliptic curves

Weil pairing

• Non-degenerate, alternating bilinear pairing $e_n: E[n] \times E[n] \to \mu_n$
• Takes values in nth roots of unity $\mu_n$
• Galois-equivariant and compatible with isogenies
• Essential tool for studying Galois representations attached to elliptic curves

Galois representations

• Action of absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on torsion points
• Yields continuous representations $\rho_n: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}/n\mathbb{Z})$
• Contain deep arithmetic information about the elliptic curve (modularity theorem)
• Crucial for studying L-functions and arithmetic properties of elliptic curves

Complex tori in higher dimensions

• Generalize one-dimensional complex tori to higher-dimensional analogues
• Provide setting for studying arithmetic of curves of higher genus
• Illustrate deep connections between complex geometry and arithmetic algebraic geometry

Abelian varieties

• Higher-dimensional analogues of elliptic curves, defined over arbitrary fields
• Complex abelian varieties isomorphic to quotients $\mathbb{C}^g/\Lambda$ for full-rank lattices $\Lambda \subset \mathbb{C}^g$
• Possess rich theory of polarizations, isogenies, and endomorphisms
• Crucial objects in arithmetic geometry, appearing in Birch and Swinnerton-Dyer conjecture

Jacobians of curves

• Abelian varieties associated to algebraic curves of any genus
• For curve $C$ of genus $g$, Jacobian $J(C)$ has dimension $g$
• Parametrize degree zero divisor classes on the curve
• Provide bridge between geometry of curves and arithmetic of abelian varieties

Poincaré reducibility theorem

• States every abelian variety is isogenous to a product of simple abelian varieties
• Simple abelian varieties have no proper abelian subvarieties
• Analogous to primary decomposition in commutative algebra
• Fundamental structure theorem for studying abelian varieties and their endomorphism rings