Elliptic curves are fascinating mathematical objects with deep connections to number theory, algebraic geometry, and cryptography. They're defined by specific equations and have a unique group structure that makes them powerful tools in various fields.

The modular j-invariant is a crucial concept in elliptic curve theory. It helps classify curves, connects them to complex analysis, and plays a key role in understanding their properties over different number fields. This chapter explores these ideas in depth.

Definition of elliptic curves

• Elliptic curves are a fundamental object of study in number theory and algebraic geometry
• They have a rich structure and deep connections to various areas of mathematics, including complex analysis, modular forms, and cryptography
• Understanding the basic definitions and properties of elliptic curves is crucial for exploring their applications and advanced theory

Weierstrass equations

• 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 convenient algebraic representation of elliptic curves
• The coefficients $a$ and $b$ determine the shape and properties of the curve
• The discriminant $\Delta = -16(4a^3 + 27b^2)$ must be nonzero for the curve to be smooth

Smooth projective curves

• Elliptic curves are smooth projective curves of genus one with a distinguished point called the point at infinity
• Smoothness means that the curve has no singularities or self-intersections
• Projective curves are defined in projective space, which allows for the inclusion of points at infinity
• The genus of a curve is a measure of its complexity and determines many of its properties

Rational points on elliptic curves

• Rational points on an elliptic curve are points whose coordinates are rational numbers
• The set of rational points on an elliptic curve forms a group under a suitable addition law
• Studying the structure and properties of the group of rational points is a central problem in elliptic curve theory
• The Mordell-Weil theorem states that the group of rational points is finitely generated

Group law on elliptic curves

• The set of points on an elliptic curve, including the point at infinity, forms an abelian group under a well-defined addition operation called the group law
• The group law endows elliptic curves with a rich algebraic structure that is essential for their applications and theoretical study
• Understanding the geometric and algebraic properties of the group law is crucial for working with elliptic curves

Geometric definition of group law

• The group law on an elliptic curve can be defined geometrically using the chord-and-tangent process
• To add two points $P$ and $Q$ on the curve, draw a line through $P$ and $Q$ (or the tangent line if $P = Q$) and find the third point of intersection with the curve, then reflect this point across the $x$-axis to obtain the sum $P + Q$
• The point at infinity serves as the identity element of the group
• The geometric definition provides a visual and intuitive understanding of the group law

Algebraic formulas for group law

• The group law on an elliptic curve can also be expressed using algebraic formulas in terms of the coordinates of the points
• For points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$, the sum $P + Q = (x_3, y_3)$ is given by:
  • $x_3 = \lambda^2 - x_1 - x_2$
  • $y_3 = \lambda(x_1 - x_3) - y_1$
  • where $\lambda = \frac{y_2 - y_1}{x_2 - x_1}$ if $P \neq Q$, and $\lambda = \frac{3x_1^2 + a}{2y_1}$ if $P = Q$
• The algebraic formulas allow for efficient computation of the group law and are used in practical implementations

Associativity and identity element

• The group law on an elliptic curve satisfies the associativity property: $(P + Q) + R = P + (Q + R)$ for any points $P$, $Q$, and $R$ on the curve
• The point at infinity, denoted by $\mathcal{O}$, serves as the identity element of the group: $P + \mathcal{O} = P$ for any point $P$ on the curve
• The inverse of a point $P = (x, y)$ is given by $-P = (x, -y)$, satisfying $P + (-P) = \mathcal{O}$

Torsion points and subgroups

• Torsion points on an elliptic curve are points of finite order, meaning that repeated addition of the point to itself eventually yields the identity element
• The set of torsion points forms a subgroup of the group of rational points on the curve
• The structure of the torsion subgroup provides valuable information about the arithmetic properties of the elliptic curve
• The Nagell-Lutz theorem gives a criterion for determining the torsion points on an elliptic curve

Isomorphism classes of elliptic curves

• Elliptic curves can be classified up to isomorphism, which means identifying curves that have the same underlying structure and properties
• Isomorphism classes provide a way to study elliptic curves in a more abstract and unified manner
• Understanding isomorphisms and the properties that remain invariant under isomorphisms is important for the classification and study of elliptic curves

Isomorphisms of elliptic curves

• Two elliptic curves are said to be isomorphic if there exists a rational function (an algebraic map) that establishes a one-to-one correspondence between their points, preserving the group law
• Isomorphisms of elliptic curves are algebraic maps of the form $(x, y) \mapsto (u^2x + r, u^3y + u^2sx + t)$, where $u, r, s, t$ are constants satisfying certain conditions
• Isomorphic curves have the same j-invariant and share many arithmetic and geometric properties

Short Weierstrass form

• Every elliptic curve is isomorphic to a curve in short Weierstrass form, given by the equation $y^2 = x^3 + ax + b$
• The short Weierstrass form provides a canonical representation of elliptic curves up to isomorphism
• Transforming an elliptic curve into short Weierstrass form simplifies its equation and facilitates the study of its properties
• The coefficients $a$ and $b$ in the short Weierstrass form are related to the invariants of the curve

j-invariant of elliptic curves

• The j-invariant is a fundamental invariant associated with an elliptic curve that characterizes its isomorphism class
• Two elliptic curves are isomorphic if and only if they have the same j-invariant
• The j-invariant is defined in terms of the coefficients of the elliptic curve and remains invariant under isomorphisms
• The j-invariant encodes important information about the structure and properties of the elliptic curve

Curves with same j-invariant

• Elliptic curves with the same j-invariant belong to the same isomorphism class and share many properties
• Curves with the same j-invariant have isomorphic torsion subgroups and similar Galois representations
• Studying families of elliptic curves with the same j-invariant can provide insights into their arithmetic and geometric behavior
• The moduli space of elliptic curves parameterizes isomorphism classes of elliptic curves and is closely related to the j-invariant

Modular j-invariant

• The modular j-invariant is a fundamental object in the theory of elliptic curves and modular forms
• It plays a central role in the study of elliptic curves over the complex numbers and their relationship to lattices and modular functions
• Understanding the properties and significance of the j-invariant is crucial for exploring the deep connections between elliptic curves, complex analysis, and number theory

Definition of j-invariant

• The j-invariant of an elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 + ax + b$ is defined as:
  • $j(E) = 1728 \frac{4a^3}{4a^3 + 27b^2}$
• The j-invariant is a rational function of the coefficients $a$ and $b$ and remains invariant under isomorphisms of elliptic curves
• The j-invariant characterizes the isomorphism class of an elliptic curve: two curves are isomorphic if and only if they have the same j-invariant

j-invariant in terms of coefficients

• The j-invariant can be expressed in terms of the coefficients of the elliptic curve in various forms
• For a curve in short Weierstrass form $y^2 = x^3 + ax + b$, the j-invariant is given by:
  • $j = 1728 \frac{4a^3}{4a^3 + 27b^2}$
• In terms of the discriminant $\Delta$ and the invariants $c_4$ and $c_6$, the j-invariant can be written as:
  • $j = \frac{c_4^3}{\Delta}$
• These expressions highlight the relationship between the j-invariant and the fundamental invariants of the elliptic curve

Properties of j-invariant

• The j-invariant satisfies several important properties that make it a key object in the study of elliptic curves:
  • It is invariant under isomorphisms of elliptic curves
  • It characterizes the isomorphism class of an elliptic curve
  • It is a modular function with respect to the action of the modular group $\text{SL}_2(\mathbb{Z})$ on the upper half-plane
• The j-invariant has a pole at the cusp (i.e., at infinity) and takes on every complex value precisely once in the fundamental domain of the modular group

j-invariant as modular function

• The j-invariant can be viewed as a modular function, meaning that it is invariant under the action of the modular group $\text{SL}_2(\mathbb{Z})$ on the upper half-plane
• As a modular function, the j-invariant satisfies certain transformation properties under the generators of the modular group
• The modular properties of the j-invariant are closely related to the theory of modular forms and elliptic modular curves
• The j-invariant provides a bridge between the arithmetic of elliptic curves and the analytic theory of modular forms

Elliptic curves over complex numbers

• Elliptic curves over the complex numbers have a rich structure and are closely connected to the theory of lattices and complex analysis
• Studying elliptic curves over $\mathbb{C}$ provides insights into their geometric and analytic properties and reveals deep connections to modular forms and complex multiplication
• The complex case serves as a foundation for understanding elliptic curves over other fields and their arithmetic behavior

Lattices and elliptic curves

• Every elliptic curve over $\mathbb{C}$ can be associated with a lattice $\Lambda$ in the complex plane, which is a discrete subgroup of $\mathbb{C}$ of rank 2
• The lattice $\Lambda$ determines the elliptic curve up to isomorphism, and conversely, every lattice gives rise to an elliptic curve
• The Weierstrass $\wp$-function associated with the lattice $\Lambda$ provides a parametrization of the elliptic curve and satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$, where $g_2$ and $g_3$ are invariants of the lattice

Periods and fundamental parallelogram

• The periods of an elliptic curve over $\mathbb{C}$ are the generators $\omega_1$ and $\omega_2$ of the associated lattice $\Lambda$
• The periods determine the shape and size of the fundamental parallelogram, which is a parallelogram in the complex plane with vertices $0$, $\omega_1$, $\omega_2$, and $\omega_1 + \omega_2$
• The ratio $\tau = \frac{\omega_2}{\omega_1}$ is called the modular parameter and lies in the upper half-plane $\mathbb{H}$
• The modular parameter $\tau$ determines the isomorphism class of the elliptic curve and is related to the j-invariant by a modular function

Uniformization theorem

• The uniformization theorem states that every elliptic curve over $\mathbb{C}$ is analytically isomorphic to a quotient of the complex plane by a lattice
• The isomorphism is given by the Weierstrass $\wp$-function and its derivative, which provide a bijection between the complex plane modulo the lattice and the points on the elliptic curve
• The uniformization theorem establishes a deep connection between elliptic curves, complex analysis, and the geometry of lattices
• It allows for the study of elliptic curves using the tools and techniques of complex function theory

Complex multiplication

• Complex multiplication refers to the phenomenon where the endomorphism ring of an elliptic curve over $\mathbb{C}$ is larger than the integers
• Elliptic curves with complex multiplication have additional symmetries and special arithmetic properties
• The complex multiplication points on the modular curve parametrize elliptic curves with complex multiplication
• The theory of complex multiplication plays a crucial role in the study of elliptic curves over number fields and their Galois representations

Elliptic curves over finite fields

• Elliptic curves over finite fields have important applications in cryptography and number theory
• The study of elliptic curves over finite fields involves understanding their reduction modulo primes, point counting, and the distinction between supersingular and ordinary curves
• The arithmetic and geometric properties of elliptic curves over finite fields are closely related to their j-invariants and endomorphism rings

Reduction of elliptic curves mod p

• Given an elliptic curve defined over the rational numbers (or a number field), its reduction modulo a prime $p$ is the curve obtained by reducing the coefficients of the Weierstrass equation modulo $p$
• The reduction of an elliptic curve modulo $p$ can be smooth (if the discriminant is nonzero modulo $p$) or singular (if the discriminant is zero modulo $p$)
• The type of reduction (good, bad, or additive) provides information about the structure of the group of points on the reduced curve
• Studying the reduction of elliptic curves modulo primes is important for understanding their arithmetic properties and Galois representations

Number of points on elliptic curves

• The number of points on an elliptic curve over a finite field $\mathbb{F}_q$ (including the point at infinity) is denoted by $#E(\mathbb{F}_q)$
• The Hasse-Weil bound states that the number of points satisfies the inequality $|#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$
• The number of points on an elliptic curve over $\mathbb{F}_q$ is related to its trace of Frobenius, which is defined as $a_q = q + 1 - #E(\mathbb{F}_q)$
• Counting the number of points on elliptic curves over finite fields is a fundamental problem with applications in cryptography and coding theory

Hasse's theorem on point counts

• Hasse's theorem provides a precise characterization of the possible values for the number of points on an elliptic curve over a finite field
• It states that for an elliptic curve $E$ over $\mathbb{F}_q$, the trace of Frobenius $a_q$ satisfies $|a_q| \leq 2\sqrt{q}$
• Equivalently, the number of points $#E(\mathbb{F}_q)$ satisfies the Hasse-Weil bound $|#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$
• Hasse's theorem imposes strong constraints on the possible point counts of elliptic curves over finite fields and is a key result in the arithmetic theory of elliptic curves

Supersingular vs ordinary curves

• Elliptic curves over finite fields can be classified as either supersingular or ordinary based on their endomorphism rings and j-invariants
• A curve $E$ over $\mathbb{F}_q$ is supersingular if its trace of Frobenius satisfies $a_q \equiv 0 \pmod{p}$, where $q = p^n$ for some prime $p$ and integer $n$
• Supersingular curves have a larger endomorphism ring than ordinary curves and have special properties that make them useful in certain cryptographic protocols
• Ordinary curves are those that are not supersingular and have endomorphism rings that are orders in imaginary quadratic fields
• The distinction between supersingular and ordinary curves is important for understanding their arithmetic and geometric properties

Applications of j-invariant

• The j-invariant of elliptic curves has numerous applications in various areas of mathematics, including complex analysis, algebraic geometry, and number theory
• It serves as a key tool for classifying elliptic curves, studying their moduli spaces, and investigating their connections