6.3 Elliptic curves and the Riemann-Roch theorem

Elliptic curves are smooth, projective algebraic curves of genus one with a specified basepoint. They have a rich structure that makes them useful in number theory, cryptography, and algebraic geometry.

The Riemann-Roch theorem is a fundamental result for elliptic curves. It relates the dimension of rational function spaces to divisor degrees, providing insights into the curve's geometry and arithmetic properties.

Definition of elliptic curves

• Elliptic curves are smooth, projective, algebraic curves of genus one with a specified basepoint
• They have a rich geometric and algebraic structure that makes them useful in various fields of mathematics, including number theory, cryptography, and algebraic geometry

Weierstrass equations

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• Elliptic curves can be defined by Weierstrass equations of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants satisfying certain conditions
• The Weierstrass equation provides a canonical form for representing elliptic curves
• The discriminant $\Delta = -16(4a^3 + 27b^2)$ must be nonzero for the curve to be smooth

Group law on elliptic curves

• Elliptic curves have a natural group structure, where the group operation is defined geometrically
• The group law allows for the addition of points on the elliptic curve
• The group law is associative, has an identity element (the point at infinity), and every point has an inverse

Elliptic curves over finite fields

• Elliptic curves can be defined over finite fields $\mathbb{F}_q$, where $q$ is a prime power
• The group of rational points on an elliptic curve over a finite field is a finite abelian group
• Elliptic curves over finite fields have applications in cryptography (elliptic curve cryptography) due to their discrete logarithm problem being hard to solve

Divisors on elliptic curves

• Divisors are formal sums of points on an elliptic curve, used to study the arithmetic and geometry of the curve
• They provide a way to describe the zeros and poles of rational functions and differentials on the curve

Definition of divisors

• A divisor on an elliptic curve $E$ is a formal sum $D = \sum n_P P$, where $P$ runs over the points of $E$ and $n_P$ are integers, with only finitely many $n_P$ nonzero
• The set of divisors on $E$ forms an abelian group under pointwise addition

Degree of divisors

• The degree of a divisor $D = \sum n_P P$ is defined as $\deg(D) = \sum n_P$
• The degree map is a homomorphism from the group of divisors to the integers
• The set of divisors of degree zero is a subgroup of the divisor group

Principal divisors

• A principal divisor is a divisor of the form $\text{div}(f) = \sum \text{ord}_P(f) P$, where $f$ is a nonzero rational function on the elliptic curve and $\text{ord}_P(f)$ is the order of vanishing of $f$ at $P$
• Principal divisors have degree zero and form a subgroup of the divisor group

Divisor class group

• The divisor class group (or Picard group) of an elliptic curve is the quotient group of the divisor group by the subgroup of principal divisors
• Two divisors are linearly equivalent if their difference is a principal divisor
• The divisor class group is a finite abelian group that encodes important information about the elliptic curve

Riemann-Roch theorem for elliptic curves

• The Riemann-Roch theorem is a fundamental result in the theory of algebraic curves, relating the dimension of the space of rational functions with prescribed poles to the degree of a divisor

Statement of Riemann-Roch theorem

• For an elliptic curve $E$ and a divisor $D$ on $E$, the Riemann-Roch theorem states that $\dim L(D) = \deg(D) + 1 - g + \dim L(K - D)$, where:
  • $L(D)$ is the Riemann-Roch space associated to $D$
  • $g$ is the genus of the curve (which is 1 for elliptic curves)
  • $K$ is a canonical divisor

Proof of Riemann-Roch theorem

• The proof of the Riemann-Roch theorem for elliptic curves relies on the properties of the divisor class group and the Serre duality theorem
• Key steps involve showing the invariance of the Euler characteristic $\chi(D) = \dim L(D) - \dim L(K - D)$ and using the fact that the canonical divisor has degree $2g-2$

Applications of Riemann-Roch theorem

• The Riemann-Roch theorem allows for the computation of the dimension of Riemann-Roch spaces
• It provides a way to study the behavior of rational functions and differentials on elliptic curves
• The theorem has applications in the theory of algebraic curves, algebraic geometry, and number theory (Mordell-Weil theorem)

Rational functions on elliptic curves

• Rational functions on elliptic curves are important objects in the study of the curve's geometry and arithmetic
• They are used to describe maps between elliptic curves and to study the divisor class group

Definition of rational functions

• A rational function on an elliptic curve $E$ is a function $f: E \to \mathbb{P}^1$ that can be written as the ratio of two polynomials in the coordinates of the curve
• The set of rational functions on $E$ forms a field, denoted by $K(E)$

Poles and zeros of rational functions

• The zeros of a rational function $f$ are the points $P$ on the elliptic curve where $f(P) = 0$
• The poles of $f$ are the points $P$ where $f$ has a singularity (i.e., $f(P) = \infty$)
• The order of a zero or pole is the multiplicity of the point in the divisor of $f$

Riemann-Roch spaces

• For a divisor $D$ on an elliptic curve $E$, the Riemann-Roch space $L(D)$ is the vector space of rational functions $f$ on $E$ such that $\text{div}(f) + D \geq 0$
• The dimension of $L(D)$ is given by the Riemann-Roch theorem
• Riemann-Roch spaces are key objects in the study of the geometry of elliptic curves

Canonical divisors

• Canonical divisors play a central role in the theory of algebraic curves and the Riemann-Roch theorem
• They are closely related to the concept of differentials on the curve

Definition of canonical divisors

• A canonical divisor on an elliptic curve $E$ is a divisor $K$ such that $L(K)$ is the space of holomorphic differentials on $E$
• On an elliptic curve, any divisor of the form $K = (P) - (O)$, where $P$ is a point on $E$ and $O$ is the point at infinity, is a canonical divisor

Degree of canonical divisors

• The degree of a canonical divisor on an elliptic curve is always equal to $2g-2$, where $g$ is the genus of the curve
• For elliptic curves, the genus is 1, so the degree of a canonical divisor is 0

Canonical class

• The canonical class of an elliptic curve is the linear equivalence class of canonical divisors
• All canonical divisors on an elliptic curve are linearly equivalent
• The canonical class is an important invariant of the elliptic curve

Differentials on elliptic curves

• Differentials on elliptic curves are closely related to the concept of canonical divisors and play a role in the Riemann-Roch theorem
• They provide a way to study the geometry and arithmetic of the curve

Definition of differentials

• A differential on an elliptic curve $E$ is a global section of the cotangent bundle of $E$
• In terms of the Weierstrass equation, a differential can be written as $\omega = dx/(2y)$

Holomorphic vs meromorphic differentials

• A holomorphic differential on an elliptic curve is a differential that has no poles
• A meromorphic differential is a differential that may have poles
• On an elliptic curve, there is a unique (up to scalar multiplication) holomorphic differential

Riemann-Roch theorem for differentials

• The Riemann-Roch theorem can be formulated in terms of differentials
• For a divisor $D$ on an elliptic curve $E$, the theorem states that $\dim \Omega(D) = \deg(D) - g + 1 + \dim L(K - D)$, where:
  • $\Omega(D)$ is the space of meromorphic differentials $\omega$ such that $\text{div}(\omega) \geq -D$
  • $g$ is the genus of the curve (which is 1 for elliptic curves)
  • $K$ is a canonical divisor

Elliptic curves over C

• The study of elliptic curves over the complex numbers $\mathbb{C}$ provides a rich interplay between complex analysis, algebraic geometry, and number theory
• Many properties of elliptic curves can be understood by studying their complex analytic structure

Lattices and elliptic curves

• An elliptic curve over $\mathbb{C}$ can be realized as the quotient of $\mathbb{C}$ by a lattice $\Lambda$
• A lattice is a discrete subgroup of $\mathbb{C}$ of the form $\Lambda = \{m\omega_1 + n\omega_2 : m, n \in \mathbb{Z}\}$, where $\omega_1$ and $\omega_2$ are complex numbers linearly independent over $\mathbb{R}$
• The Weierstrass $\wp$-function associated to a lattice $\Lambda$ provides a parametrization of the elliptic curve $\mathbb{C}/\Lambda$

Isomorphisms between elliptic curves

• Two elliptic curves over $\mathbb{C}$ are isomorphic if and only if their associated lattices are homothetic (i.e., one is a scalar multiple of the other)
• The j-invariant of an elliptic curve is a complex number that characterizes the isomorphism class of the curve
• Elliptic curves with the same j-invariant are isomorphic over $\mathbb{C}$

Complex tori and elliptic curves

• A complex torus is a quotient of $\mathbb{C}^n$ by a lattice of rank $2n$
• Elliptic curves over $\mathbb{C}$ are precisely the complex tori of dimension 1
• The study of complex tori and their morphisms is a central topic in complex algebraic geometry

Elliptic curves over Q

• The study of elliptic curves over the rational numbers $\mathbb{Q}$ is of great interest in number theory
• Many Diophantine equations can be reformulated as questions about rational points on elliptic curves

Mordell-Weil theorem

• The Mordell-Weil theorem states that the group of rational points on an elliptic curve over $\mathbb{Q}$ is finitely generated
• The group $E(\mathbb{Q})$ is isomorphic to $\mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}$, where $r$ is the rank of the curve and $E(\mathbb{Q})_{\text{tors}}$ is the torsion subgroup
• Computing the rank of an elliptic curve is a difficult problem, and there is no known algorithm for determining it in general

Torsion points on elliptic curves

• A torsion point on an elliptic curve $E$ over $\mathbb{Q}$ is a point of finite order in the group $E(\mathbb{Q})$
• The torsion subgroup $E(\mathbb{Q})_{\text{tors}}$ is always finite, and its possible structures are known (Mazur's theorem)
• Torsion points play a role in the study of rational points on elliptic curves and in applications to cryptography

Height functions and descent

• Height functions are tools used to measure the "size" of rational points on an elliptic curve
• The canonical height is a quadratic form on $E(\mathbb{Q})$ that behaves well under the group law and can be used to prove the Mordell-Weil theorem
• Descent is a method for computing the rank of an elliptic curve by studying the curve's torsion points and isogenies
• The method of descent provides a way to bound the rank of an elliptic curve and to search for rational points