7.3 Elliptic curves and the Taniyama-Shimura conjecture

Elliptic curves are fascinating mathematical objects with deep connections to number theory and cryptography. They're defined by specific equations and have a unique group structure, making them powerful tools for solving complex problems.

The Taniyama-Shimura conjecture, now known as the modularity theorem, links elliptic curves to modular forms. This connection was crucial in proving Fermat's Last Theorem and continues to drive research in number theory and related fields.

Elliptic curves

• Elliptic curves are a fundamental object of study in number theory and algebraic geometry
• They have important applications in cryptography and the proof of Fermat's Last Theorem
• Understanding the properties and structure of elliptic curves is crucial for this course

Definition of elliptic curves

Top images from around the web for Definition of elliptic curves

• 

  The Math Behind Elliptic Curves in Weierstrass Form - Sefik Ilkin Serengil View original

  Is this image relevant?

• 

  A simple Elliptic Curve View original

  Is this image relevant?

• 

  Annotated curve E with points P, Q, R, R' and line L labeled. View original

  Is this image relevant?

• 

  The Math Behind Elliptic Curves in Weierstrass Form - Sefik Ilkin Serengil View original

  Is this image relevant?

• 

  A simple Elliptic Curve View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of elliptic curves

• 

  The Math Behind Elliptic Curves in Weierstrass Form - Sefik Ilkin Serengil View original

  Is this image relevant?

• 

  A simple Elliptic Curve View original

  Is this image relevant?

• 

  Annotated curve E with points P, Q, R, R' and line L labeled. View original

  Is this image relevant?

• 

  The Math Behind Elliptic Curves in Weierstrass Form - Sefik Ilkin Serengil View original

  Is this image relevant?

• 

  A simple Elliptic Curve View original

  Is this image relevant?

1 of 3

• An elliptic curve is a smooth, projective algebraic curve of genus one with a specified base point
• Can be defined over any field (rational numbers, real numbers, complex numbers, finite fields)
• Non-singular cubic curve in the projective plane

Weierstrass equation

• Every elliptic curve can be described by a Weierstrass equation of the form $y^2 = x^3 + ax + b$
  • $a$ and $b$ are constants satisfying certain conditions to ensure smoothness
• Provides a standard way to represent and study elliptic curves
• Coefficients of the Weierstrass equation determine the specific shape and properties of the curve

Group law on elliptic curves

• Elliptic curves have a natural group structure defined geometrically
• Three collinear points on the curve (counting multiplicity) sum to the identity element (point at infinity)
• Group law allows for point addition and scalar multiplication
  • Enables the use of elliptic curves in cryptographic protocols (ECDH, ECDSA)

Elliptic curves over finite fields

• Elliptic curves can be defined over finite fields $\mathbb{F}_q$, where $q$ is a prime power
• Number of points on an elliptic curve over a finite field is finite and satisfies Hasse's theorem
  • |#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}
• Elliptic curve discrete logarithm problem (ECDLP) is believed to be hard, providing security for cryptographic applications

Elliptic curve cryptography

• Elliptic curves over finite fields are used in various cryptographic protocols
  • Elliptic Curve Diffie-Hellman (ECDH) for key exchange
  • Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signatures
• Offers similar security to traditional public-key cryptography (RSA) with smaller key sizes
• Widely used in modern cryptographic applications (TLS, Bitcoin, etc.)

Modularity theorem

• The modularity theorem, previously known as the Taniyama-Shimura conjecture, establishes a profound connection between elliptic curves and modular forms
• It has significant consequences in number theory, including the proof of Fermat's Last Theorem
• Understanding the modularity theorem is essential for advanced studies in elliptic curves and related areas

Statement of Taniyama-Shimura conjecture

• Every elliptic curve over the rational numbers is modular
  • Associated with a unique modular form of a specific level and weight
• Connects two seemingly distinct areas of mathematics: elliptic curves and modular forms
• Conjecture was posed in the 1950s and became a central problem in number theory

Elliptic curves and modular forms

• Modular forms are complex analytic functions with certain symmetry properties
  • Defined on the upper half-plane and invariant under the action of congruence subgroups of $\text{SL}_2(\mathbb{Z})$
• Modularity theorem states that every elliptic curve over $\mathbb{Q}$ has an associated modular form
  • L-function of the elliptic curve equals the L-function of the corresponding modular form

Frey curves and 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 $n > 2$
• In 1980s, Gerhard Frey constructed an elliptic curve (Frey curve) from a hypothetical counterexample to Fermat's Last Theorem
  • Frey curve would have unusual properties incompatible with the modularity theorem

Wiles' proof of Fermat's last theorem

• Andrew Wiles, building on work by Richard Taylor, proved the modularity theorem for semistable elliptic curves in 1995
  • Sufficient to prove Fermat's Last Theorem by contradiction using Frey curves
• Wiles' proof was a landmark achievement in number theory and relied on deep ideas from elliptic curves and modular forms

Consequences of modularity theorem

• Modularity theorem has numerous consequences beyond Fermat's Last Theorem
  • Provides a powerful tool for studying the arithmetic of elliptic curves
  • Enables the computation of elliptic curve L-functions and the investigation of their properties
• Modularity theorem has been generalized to other settings (e.g., elliptic curves over totally real fields)
  • Continues to be an active area of research in number theory

L-functions

• L-functions are complex analytic functions associated with various mathematical objects, including elliptic curves
• They encode important arithmetic and geometric information about the underlying object
• L-functions play a central role in the study of elliptic curves and are related to deep conjectures in number theory

Definition of L-functions

• An L-function is a complex analytic function defined as an Euler product over primes
  • $L(s) = \prod_p L_p(s)$, where $L_p(s)$ are local factors depending on the prime $p$
• L-functions are defined for various objects: elliptic curves, modular forms, number fields, etc.
• Convergence and analytic continuation of L-functions are important questions in analytic number theory

L-functions of elliptic curves

• Every elliptic curve $E$ over a number field has an associated L-function $L(E, s)$
  • Defined using the local zeta functions of $E$ at each prime
• L-function encodes arithmetic information about the elliptic curve
  • Coefficients of the L-function are related to the number of points on $E$ over finite fields

Hasse-Weil conjecture

• The Hasse-Weil conjecture states that the L-function of an elliptic curve has an analytic continuation and satisfies a functional equation
  • Relates values of $L(E, s)$ at $s$ and $2-s$
• Modularity theorem implies the Hasse-Weil conjecture for elliptic curves over $\mathbb{Q}$
  • Follows from the corresponding properties of modular form L-functions

Birch and Swinnerton-Dyer conjecture

• The Birch and Swinnerton-Dyer (BSD) conjecture relates the rank of an elliptic curve to the behavior of its L-function at $s=1$
  • Rank of $E$ equals the order of vanishing of $L(E, s)$ at $s=1$
• BSD conjecture also predicts the leading coefficient of the Taylor expansion of $L(E, s)$ at $s=1$
  • Involves various arithmetic invariants of the elliptic curve (Tate-Shafarevich group, regulator, etc.)

Applications of L-functions

• L-functions have numerous applications in the study of elliptic curves and beyond
  • BSD conjecture provides a way to compute the rank of an elliptic curve
  • Special values of L-functions are related to important arithmetic quantities (e.g., Birch-Swinnerton-Dyer formula)
• L-functions are also used in the study of other objects (modular forms, number fields, etc.)
  • Provide a unifying framework for various problems in number theory

Galois representations

• Galois representations are a fundamental tool in modern number theory, connecting arithmetic and geometric properties of objects
• They provide a way to study the action of Galois groups on various structures, including elliptic curves
• Galois representations are closely related to modularity and have important applications in the theory of elliptic curves

Galois groups and representations

• The Galois group $G_K$ of a field extension $K/\mathbb{Q}$ is the group of automorphisms of $K$ fixing $\mathbb{Q}$
• A Galois representation is a continuous homomorphism $\rho: G_K \to \text{GL}_n(R)$, where $R$ is a topological ring
  • Encodes the action of $G_K$ on an $n$-dimensional vector space over $R$
• Galois representations arise naturally in various contexts (étale cohomology, Tate modules, etc.)

Tate modules of elliptic curves

• The Tate module $T_\ell(E)$ of an elliptic curve $E$ over a field $K$ is the inverse limit of the $\ell$-power torsion points of $E$
  • $T_\ell(E) = \varprojlim E[\ell^n]$, where $\ell$ is a prime different from the characteristic of $K$
• Tate module is a free $\mathbb{Z}_\ell$-module of rank 2, equipped with a continuous action of $G_K$
  • Gives rise to a Galois representation $\rho_{E,\ell}: G_K \to \text{GL}_2(\mathbb{Z}_\ell)$

Galois representations attached to elliptic curves

• The Tate module construction provides a way to attach a Galois representation to an elliptic curve
• Properties of the elliptic curve are reflected in the properties of the associated Galois representation
  • Reduction type of $E$ at primes related to the image of Frobenius elements under $\rho_{E,\ell}$
• Studying the Galois representations of elliptic curves is a powerful tool in understanding their arithmetic

Serre's conjecture

• Serre's conjecture, now a theorem (Khare-Wintenberger, Kisin), describes the Galois representations arising from modular forms
  • Every irreducible, odd, continuous Galois representation $\rho: G_\mathbb{Q} \to \text{GL}_2(\mathbb{F}_p)$ comes from a modular form
• Serre's conjecture provides a converse to the modularity theorem for Galois representations
  • Establishes a bijection between certain Galois representations and modular forms

Modularity and Galois representations

• The modularity theorem has important consequences for Galois representations of elliptic curves
  • Galois representation $\rho_{E,\ell}$ of a modular elliptic curve $E$ is isomorphic to the Galois representation attached to the corresponding modular form
• Modularity allows for the study of elliptic curve Galois representations using the tools and techniques of modular forms
  • Provides a rich interplay between the arithmetic of elliptic curves and the theory of modular forms

Elliptic surfaces

• Elliptic surfaces are a natural generalization of elliptic curves, combining aspects of algebraic geometry and complex geometry
• They provide a geometric framework for studying families of elliptic curves and their variations
• The theory of elliptic surfaces has important applications in the study of rational points and Diophantine equations

Elliptic fibrations

• An elliptic surface is a complex surface $S$ equipped with a morphism $f: S \to C$ to a smooth curve $C$, such that almost all fibers are smooth elliptic curves
  • $f$ is called an elliptic fibration
• Elliptic surfaces can be described locally by Weierstrass equations with coefficients that are functions on the base curve $C$
  • Coefficients determine the specific elliptic curve over each point of $C$

Kodaira-Néron classification

• The Kodaira-Néron classification describes the possible types of singular fibers in an elliptic fibration
  • Singular fibers are classified into types $I_n$, $II$, $III$, $IV$, $I_n^*$, $II^*$, $III^*$, $IV^*$
• Each type corresponds to a specific configuration of the elliptic curve degenerating over a point of the base curve
  • Classification is based on the local monodromy and the structure of the special fiber

Singular fibers and Euler characteristics

• The singular fibers of an elliptic fibration contribute to the Euler characteristic of the elliptic surface
  • Each type of singular fiber has a specific contribution, given by the Ogg-Shafarevich formula
• The Euler characteristic is a topological invariant that measures the complexity of the elliptic surface
  • Related to the geometry of the base curve and the singular fibers

Mordell-Weil theorem for elliptic surfaces

• The Mordell-Weil theorem states that the group of rational points (sections) of an elliptic surface over a number field is finitely generated
  • Generalizes the Mordell-Weil theorem for elliptic curves
• The rank of the Mordell-Weil group measures the number of independent rational sections
  • Relates to the geometry of the elliptic surface and the base curve

Rational points on elliptic surfaces

• The study of rational points on elliptic surfaces is a central problem in Diophantine geometry
  • Relates to questions about the existence and density of rational points on the fibers (elliptic curves)
• Techniques from the theory of elliptic surfaces (Kodaira-Néron classification, Mordell-Weil theorem, etc.) are used to study rational points
  • Provides a geometric approach to Diophantine problems
• The distribution of rational points on elliptic surfaces is a topic of ongoing research, with connections to arithmetic geometry and number theory