7.6 Elliptic curves and the Modularity theorem

Elliptic curves are fascinating mathematical objects with deep connections to number theory and cryptography. They're defined by equations like y^2 = x^3 + ax + b and form abelian groups under a geometric addition law. This structure makes them powerful tools in various fields.

The Modularity Theorem links elliptic curves over rational numbers to modular forms, complex analytic functions with special symmetries. This profound connection, proven in 2001, resolved Fermat's Last Theorem and opened new avenues in number theory and cryptography.

Elliptic curves over fields

• Elliptic curves are algebraic curves defined over various fields, including the real numbers, complex numbers, and finite fields
• The study of elliptic curves over fields is a fundamental aspect of algebraic geometry and number theory
• Understanding the properties and behavior of elliptic curves over different fields is crucial for applications in cryptography and solving Diophantine equations

Weierstrass equations

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• Elliptic curves can be described using Weierstrass equations of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants from the field
• The Weierstrass equation provides a standard form for representing elliptic curves, making it easier to study their properties and perform computations
• The coefficients $a$ and $b$ in the Weierstrass equation determine the shape and characteristics of the elliptic curve
• The Weierstrass equation is used to define the group law for points on the elliptic curve

Group law for points

• The set of points on an elliptic curve, together with a special point called the "point at infinity," forms an abelian group under a well-defined group law
• The group law for points on an elliptic curve is defined geometrically using the chord-and-tangent method
  • Given two points $P$ and $Q$ on the curve, draw a line through them and find the third point of intersection with the curve, then reflect that point across the $x$-axis to obtain the sum $P+Q$
  • If $P=Q$, the line is taken to be the tangent line at $P$, and the sum is $2P$
• The group law satisfies the properties of associativity, identity (point at infinity), and inverses
• The group structure of elliptic curves is the foundation for their use in cryptography and other applications

Discriminant and j-invariant

• The discriminant $\Delta$ of an elliptic curve is a quantity that determines whether the curve is smooth (non-singular) or has singularities
  • For a Weierstrass equation $y^2 = x^3 + ax + b$, the discriminant is given by $\Delta = -16(4a^3 + 27b^2)$
  • If $\Delta \neq 0$, the curve is smooth; if $\Delta = 0$, the curve has a singularity (node or cusp)
• The j-invariant is a quantity that characterizes the isomorphism class of an elliptic curve
  • Two elliptic curves are isomorphic over an algebraically closed field if and only if they have the same j-invariant
  • The j-invariant is given by $j = 1728 \frac{4a^3}{4a^3 + 27b^2}$ for a Weierstrass equation $y^2 = x^3 + ax + b$

Isomorphism classes

• Elliptic curves over a field can be classified into isomorphism classes based on their j-invariants
• Isomorphic elliptic curves have the same j-invariant and share many properties, such as the same group structure and the same number of points over finite fields
• Understanding isomorphism classes is important for studying the moduli space of elliptic curves and for applications in cryptography

Torsion points and structure

• Torsion points on an elliptic curve are points of finite order under the group law, meaning that a multiple of the point by some integer yields the point at infinity
• The set of torsion points on an elliptic curve forms a subgroup of the curve's group structure
• The torsion subgroup of an elliptic curve over a field can have different structures depending on the characteristic of the field
  • Over the complex numbers, the torsion subgroup is isomorphic to a product of two cyclic groups
  • Over the rational numbers, the torsion subgroup is either cyclic or a product of two cyclic groups with specific possible orders (Mazur's theorem)
• Understanding the torsion structure of elliptic curves is important for studying their arithmetic properties and for applications in cryptography

Elliptic curves over complex numbers

• Elliptic curves over the complex numbers have a rich structure and deep connections to complex analysis and algebraic geometry
• The study of elliptic curves over the complex numbers reveals important relationships between elliptic functions, modular forms, and lattices

Lattices and isomorphism

• Every elliptic curve over the complex numbers is isomorphic to a complex torus, which is the quotient of the complex plane by a lattice
• A lattice is a discrete subgroup of the complex plane 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 the real numbers
• The isomorphism between an elliptic curve and a complex torus is given by the Weierstrass $\wp$-function and its derivative
• The period lattice of an elliptic curve determines its isomorphism class and j-invariant

Elliptic functions and invariants

• Elliptic functions are meromorphic functions on the complex plane that are periodic with respect to a lattice
• The Weierstrass $\wp$-function is a fundamental elliptic function that 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
• The Eisenstein series $G_4$ and $G_6$ are modular forms that can be used to express the invariants $g_2$ and $g_3$ in terms of the lattice
• The j-invariant of an elliptic curve can be expressed in terms of the Eisenstein series as $j = 1728 \frac{G_4^3}{G_4^3 - 27G_6^2}$

Complex multiplication

• Some elliptic curves over the complex numbers have additional symmetries, known as complex multiplication (CM)
• An elliptic curve has complex multiplication if its endomorphism ring, the ring of isogenies from the curve to itself, is larger than the integers
• Elliptic curves with complex multiplication correspond to lattices that are invariant under certain imaginary quadratic fields
• The j-invariants of elliptic curves with complex multiplication are algebraic integers, and they play a crucial role in the theory of class fields and the construction of abelian extensions of imaginary quadratic fields

Modular curves and functions

• Modular curves are algebraic curves that parametrize isomorphism classes of elliptic curves together with additional structure, such as torsion points or isogenies
• The modular curve $X(1)$ parametrizes isomorphism classes of elliptic curves, and it is isomorphic to the complex plane quotiented by the modular group $\text{SL}(2,\mathbb{Z})$
• Modular functions are functions on the upper half-plane that are invariant under the action of subgroups of the modular group
• The j-invariant is an example of a modular function, and it provides an isomorphism between the modular curve $X(1)$ and the complex plane
• Modular curves and functions play a central role in the study of elliptic curves and their moduli spaces, and they have important applications in number theory and cryptography

Elliptic curves over finite fields

• Elliptic curves over finite fields have important applications in cryptography and coding theory
• The study of elliptic curves over finite fields involves understanding their point counts, endomorphism rings, and algorithms for efficient computation

Hasse's theorem on point counts

• Hasse's theorem provides bounds on the number of points on an elliptic curve over a finite field
• For an elliptic curve $E$ over a finite field $\mathbb{F}_q$, the number of points $\#E(\mathbb{F}_q)$ satisfies the inequality $|\#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$
• The quantity $a_q = q+1 - \#E(\mathbb{F}_q)$ is called the trace of Frobenius and plays a crucial role in the study of elliptic curves over finite fields
• Hasse's theorem is a consequence of the Riemann hypothesis for curves over finite fields, which was proved by André Weil

Supersingular vs ordinary curves

• Elliptic curves over finite fields can be classified into two types: supersingular and ordinary
• An elliptic curve $E$ over a finite field $\mathbb{F}_q$ of characteristic $p$ is called supersingular if the trace of Frobenius $a_q$ is divisible by $p$; otherwise, it is called ordinary
• Supersingular curves have special properties and are of particular interest in cryptography and coding theory
  • They have a larger endomorphism ring than ordinary curves
  • They are more resistant to certain attacks in elliptic curve cryptography
• The j-invariants of supersingular curves over a given finite field are roots of the supersingular polynomial, which has degree at most $\lfloor \frac{p}{12} \rfloor$

Endomorphism rings

• The endomorphism ring of an elliptic curve over a finite field is the ring of isogenies from the curve to itself
• For ordinary elliptic curves, the endomorphism ring is an order in an imaginary quadratic field
• For supersingular curves, the endomorphism ring is an order in a quaternion algebra
• The structure of the endomorphism ring provides information about the arithmetic and geometric properties of the elliptic curve
• Understanding the endomorphism ring is important for constructing efficient algorithms for point counting and other computations on elliptic curves over finite fields

Schoof's algorithm for point counting

• Schoof's algorithm is a polynomial-time algorithm for counting the number of points on an elliptic curve over a finite field
• The algorithm uses the action of the Frobenius endomorphism on the $\ell$-torsion points of the curve for various small primes $\ell$
• By computing the trace of Frobenius modulo several small primes and using the Chinese remainder theorem, Schoof's algorithm determines the trace of Frobenius and hence the point count
• Schoof's algorithm has a running time of $O(\log^8 q)$ for a curve over $\mathbb{F}_q$, which is polynomial in the size of the input
• Improvements to Schoof's algorithm, such as the Schoof-Elkies-Atkin (SEA) algorithm, have led to even faster point-counting methods for elliptic curves over finite fields

Modularity theorem

• The modularity theorem, also known as the Taniyama-Shimura conjecture or the Taniyama-Shimura-Weil conjecture, establishes a profound connection between elliptic curves over the rational numbers and modular forms
• The theorem states that every elliptic curve over the rational numbers is modular, meaning that it corresponds to a modular form of a specific level and weight
• The modularity theorem was initially conjectured in the 1950s and 1960s, and it was fully proved in 2001 by Breuil, Conrad, Diamond, and Taylor, building on the work of Andrew Wiles and others

Elliptic curves over rationals

• An elliptic curve over the rational numbers is a curve of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are rational numbers, and the curve has no singular points
• The set of rational points on an elliptic curve, together with the point at infinity, forms a finitely generated abelian group (Mordell-Weil theorem)
• The rank of an elliptic curve over the rationals is the number of free generators of its group of rational points
• Computing the rank of an elliptic curve is a difficult problem, and it is related to the Birch and Swinnerton-Dyer conjecture, one of the Millennium Prize Problems

Modular forms and Hecke operators

• Modular forms are complex analytic functions on the upper half-plane that satisfy certain transformation properties under the action of subgroups of the modular group $\text{SL}(2,\mathbb{Z})$
• A modular form $f$ of weight $k$ and level $N$ is a holomorphic function on the upper half-plane that satisfies $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$ for all $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$, and is holomorphic at the cusps
• Hecke operators are linear operators that act on the space of modular forms of a given weight and level
• Hecke operators are used to define Hecke eigenforms, which are simultaneous eigenfunctions of all Hecke operators
• The Fourier coefficients of a Hecke eigenform satisfy certain multiplicative properties and are related to the eigenvalues of the Hecke operators

Galois representations

• A Galois representation is a continuous homomorphism from the absolute Galois group of a field (usually the rational numbers) to the general linear group of a vector space over a field (usually a finite field or a $p$-adic field)
• Elliptic curves and modular forms both give rise to Galois representations
  • The action of the absolute Galois group on the torsion points of an elliptic curve defines a Galois representation
  • The Fourier coefficients of a Hecke eigenform generate a number field, and the action of the absolute Galois group on the roots of the characteristic polynomial of the Hecke operators defines a Galois representation
• The modularity theorem establishes a correspondence between the Galois representations arising from elliptic curves and those arising from modular forms

L-functions of elliptic curves

• The L-function of an elliptic curve over the rational numbers is a complex analytic function that encodes important arithmetic information about the curve
• The L-function of an elliptic curve $E$ is defined as a product of local factors $L_p(E,s)$ over all primes $p$, where the local factors are determined by the number of points on the curve modulo $p$
• The modularity theorem implies that the L-function of an elliptic curve over the rationals coincides with the L-function of a modular form of weight 2 and a certain level
• The Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve to the order of vanishing of its L-function at $s=1$

Frey curves and Fermat's last theorem

• The modularity theorem played a crucial role in Andrew Wiles' proof of Fermat's last theorem
• Fermat's last theorem states that the equation $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$
• Wiles' proof relied on showing that a certain class of elliptic curves, known as Frey curves, cannot exist
  • A Frey curve is an elliptic curve constructed from a hypothetical solution to Fermat's equation
  • The modularity theorem, combined with work on Galois representations and modular forms, led to a contradiction that proved the non-existence of Frey curves
• The proof of Fermat's last theorem is a testament to the power and importance of the modularity theorem in number theory

Applications of modularity

• The modularity theorem has numerous applications in number theory, including the study of Diophantine equations, congruent numbers, and the Birch and Swinnerton-Dyer conjecture
• The connection between elliptic curves and modular forms provided by the modularity theorem has led to new insights and techniques in solving long-standing problems

Congruent number problem

• A congruent number is a positive integer that is the area of a right triangle with rational side lengths
• The congruent number problem asks which integers are congruent numbers
• The congruent number problem can be reformulated in terms of the rank of elliptic curves
  • An integer $n$ is a congruent number if and only if the elliptic curve $E_n: y^2 = x^3 - n^2x$ has positive rank over the rational numbers
• The modularity theorem provides a powerful tool for studying the congruent number problem, as it relates the rank of the elliptic curve to the vanishing of the associated modular form's L-function

Sums of squares as congruent numbers

• The modularity theorem has been used to prove results about the representation of integers as sums of squares
• For example, a positive integer $n$ is a congruent number if and only if the elliptic curve $E_n: y^2 = x^3 - n^2x$ has a rational point of infinite order
• Using the modularity theorem and the theory of modular forms, it has been shown that if $n$ is a sum of two rational squares, then $E_n$ has a rational point of