Elliptic curves and complex tori are deeply connected mathematical objects. By viewing elliptic curves as quotients of the complex plane by lattices, we gain insights into their geometry and topology, bridging complex analysis and algebraic geometry.
This perspective reveals the isomorphism between elliptic curves and complex tori, established through the Weierstrass ℘-function. This connection allows us to study elliptic curves using complex analysis tools and explore their group structure, j-invariants, and applications in number theory.
Elliptic curves as complex tori
- Elliptic curves can be viewed as complex tori, which are quotient spaces of the complex plane by a lattice
- This perspective allows for a deeper understanding of the geometry and topology of elliptic curves
- Studying elliptic curves as complex tori reveals connections to other areas of mathematics, such as complex analysis and algebraic geometry
Lattices in the complex plane
- A lattice in the complex plane is a discrete subgroup of ℂ generated by two linearly independent complex numbers $\omega_1$ and $\omega_2$
- The lattice $\Lambda = {n\omega_1 + m\omega_2 : n, m \in ℤ}$ forms a regular grid in the complex plane
- The choice of generating vectors $\omega_1$ and $\omega_2$ determines the shape and orientation of the lattice (rectangular, square, hexagonal)
- The fundamental parallelogram of the lattice is the region ${a\omega_1 + b\omega_2 : 0 ≤ a, b < 1}$, which tiles the complex plane when translated by lattice vectors
Quotient spaces and tori
- The quotient space ℂ/Λ is obtained by identifying points in the complex plane that differ by a lattice vector
- ℂ/Λ has the topology of a torus (donut shape) since opposite sides of the fundamental parallelogram are identified
- The complex structure on ℂ induces a complex structure on the quotient space ℂ/Λ, making it a complex torus
- The periods of the lattice $\omega_1$ and $\omega_2$ determine the shape and size of the resulting torus
Periods of elliptic curves
- Every elliptic curve E has an associated lattice Λ_E, and the quotient space ℂ/Λ_E is isomorphic to E as a complex manifold
- The periods of an elliptic curve are the generating vectors $\omega_1$ and $\omega_2$ of its associated lattice Λ_E
- The ratio $\tau = \omega_2/\omega_1$ (in the upper half-plane) uniquely determines the isomorphism class of the elliptic curve
- The j-invariant of an elliptic curve can be expressed in terms of its periods, linking the complex analytic and algebraic perspectives
Isomorphism theorem
- The isomorphism theorem states that every elliptic curve over ℂ is isomorphic to a complex torus ℂ/Λ for some lattice Λ
- This isomorphism provides a powerful tool for studying elliptic curves using complex analysis and algebraic geometry
- The isomorphism is established using the Weierstrass ℘-function and its properties
Weierstrass ℘-function
- The Weierstrass ℘-function associated to a lattice Λ is defined as $℘(z) = \frac{1}{z^2} + \sum_{\omega \in Λ \setminus {0}} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)$
- ℘(z) is an even, doubly periodic meromorphic function with poles at the lattice points
- The ℘-function satisfies the differential equation $℘'(z)^2 = 4℘(z)^3 - g_2℘(z) - g_3$, where $g_2$ and $g_3$ are constants depending on the lattice
- The map $φ: ℂ/Λ → E, z ↦ (℘(z), ℘'(z))$ defines an isomorphism between the complex torus ℂ/Λ and the elliptic curve $E: y^2 = 4x^3 - g_2x - g_3$
Elliptic functions and meromorphic functions
- An elliptic function is a meromorphic function on ℂ that is doubly periodic with respect to a lattice Λ
- The field of elliptic functions associated to a lattice Λ is generated by the Weierstrass ℘-function and its derivative ℘'
- Every elliptic function can be expressed as a rational function in ℘ and ℘'
- The isomorphism theorem implies that the field of meromorphic functions on an elliptic curve is isomorphic to the field of elliptic functions associated to its lattice
Isomorphism between elliptic curves and tori
- The isomorphism $φ: ℂ/Λ → E$ establishes a one-to-one correspondence between points on the complex torus and points on the elliptic curve
- The isomorphism preserves the complex analytic structure, making it a biholomorphic map
- Under the isomorphism, the group structure on the elliptic curve (defined by the chord-tangent law) corresponds to the natural group structure on the torus (addition of complex numbers modulo the lattice)
- The isomorphism allows for the study of elliptic curves using the tools of complex analysis and the study of complex tori using algebraic geometry
Properties of the isomorphism
- The isomorphism between elliptic curves and complex tori has several important properties that highlight the deep connections between these objects
- These properties allow for the transfer of results and techniques between the realms of complex analysis and algebraic geometry
Isomorphism preserving group structure
- The isomorphism $φ: ℂ/Λ → E$ is a group homomorphism, preserving the additive group structure on both sides
- Addition of points on the elliptic curve E (defined by the chord-tangent law) corresponds to addition of complex numbers modulo the lattice Λ
- The identity element of the group E is the point at infinity, which corresponds to the zero element of ℂ/Λ
- Torsion points on the elliptic curve (points of finite order) correspond to division points of the lattice (complex numbers that are rational multiples of lattice vectors)
Isomorphism as a biholomorphic map
- The isomorphism $φ: ℂ/Λ → E$ is a biholomorphic map, meaning it is a bijective holomorphic map with a holomorphic inverse
- The complex analytic structure on the torus ℂ/Λ (inherited from ℂ) is preserved under the isomorphism
- Holomorphic maps between elliptic curves correspond to holomorphic maps between their associated complex tori that respect the lattice structure
- The isomorphism allows for the study of geometric properties of elliptic curves using complex analytic techniques
Isomorphism and j-invariant
- The j-invariant of an elliptic curve E is a complex number that uniquely determines its isomorphism class over ℂ
- Two elliptic curves over ℂ are isomorphic if and only if they have the same j-invariant
- The j-invariant can be expressed in terms of the periods $\omega_1$ and $\omega_2$ of the associated lattice Λ_E, linking the complex analytic and algebraic perspectives
- The j-invariant provides a way to classify elliptic curves over ℂ and study their moduli space
Consequences and applications
- The isomorphism between elliptic curves and complex tori has far-reaching consequences and applications in various areas of mathematics
- It allows for the transfer of results and techniques between complex analysis, algebraic geometry, and number theory, leading to deep insights and new discoveries
Complex multiplication of elliptic curves
- An elliptic curve E has complex multiplication if its associated lattice Λ_E is invariant under multiplication by a complex number $α ∉ ℚ$
- Elliptic curves with complex multiplication have special arithmetic properties and are important in number theory
- The j-invariants of elliptic curves with complex multiplication are algebraic integers, and their minimal polynomials have interesting properties
- The theory of complex multiplication provides a link between elliptic curves and class field theory
Elliptic curves over ℂ vs over ℚ
- The isomorphism theorem holds for elliptic curves over ℂ, but the situation is more complicated for elliptic curves over ℚ (or other fields)
- Not every elliptic curve over ℚ is isomorphic to a complex torus with a lattice defined over ℚ
- The study of elliptic curves over ℚ leads to questions about rational points, torsion subgroups, and the Mordell-Weil theorem
- The interplay between the complex analytic and algebraic aspects of elliptic curves over ℚ is a central theme in arithmetic geometry
Modular curves and moduli spaces
- Modular curves are algebraic curves that parametrize isomorphism classes of elliptic curves with additional structure (e.g., torsion points, isogenies)
- The j-invariant provides a natural map from a modular curve to the moduli space of elliptic curves (a coarse moduli space)
- The study of modular curves and their properties leads to important results in number theory, such as the proof of Fermat's Last Theorem
- The isomorphism between elliptic curves and complex tori provides a foundation for the study of moduli spaces and their compactifications