🔢Elliptic Curves Unit 7 – Elliptic curves and modular forms

Key Concepts and Definitions

• Elliptic curve defined as a smooth, projective algebraic curve of genus one with a specified basepoint
• Weierstrass equation $y^2 = x^3 + ax + b$ represents elliptic curves in short Weierstrass form
• Modular form is a complex analytic function on the upper half-plane satisfying certain transformation properties under the action of the modular group $SL(2,\mathbb{Z})$
• Modularity theorem establishes a bijective correspondence between elliptic curves over $\mathbb{Q}$ and modular forms of weight 2 and level N
• Torsion points on an elliptic curve are points of finite order under the group law
• Conductor of an elliptic curve measures the degree of bad reduction at primes
• L-function associated to an elliptic curve encodes arithmetic information and is related to the L-function of the corresponding modular form

Historical Context and Development

• Elliptic curves first studied in connection with elliptic integrals arising in the calculation of arc lengths of ellipses in the 17th and 18th centuries
• Modular forms emerged in the work of Eisenstein, Jacobi, and Riemann on complex analysis and number theory in the 19th century
• Taniyama-Shimura conjecture (1950s) proposed a link between elliptic curves and modular forms, which was later refined by Weil
• Frey curve, proposed by Gerhard Frey in the 1980s, provided a connection between Fermat's Last Theorem and elliptic curves
• Andrew Wiles, building on work of Ken Ribet, proved the modularity theorem for semistable elliptic curves in 1995, leading to the proof of Fermat's Last Theorem
  • Wiles' proof was completed by Richard Taylor and others, covering all elliptic curves over $\mathbb{Q}$
• Birch and Swinnerton-Dyer conjecture (1960s) relates the rank of an elliptic curve to the behavior of its L-function, remains unproven

Elliptic Curves: Structure and Properties

• Elliptic curves form an abelian group under the chord-and-tangent law, with the basepoint serving as the identity element
• Elliptic curves over $\mathbb{C}$ are parametrized by the j-invariant, with two curves being isomorphic if and only if they have the same j-invariant
• Elliptic curves over $\mathbb{Q}$ can be classified by their conductor, which measures the primes of bad reduction
• Mordell-Weil theorem states that the group of rational points on an elliptic curve over a number field is finitely generated
• Torsion subgroup of an elliptic curve over $\mathbb{Q}$ is finite, with possible structures classified by Mazur's torsion theorem
• Elliptic curves can be reduced modulo primes, leading to the study of elliptic curves over finite fields
  • Hasse's theorem bounds the number of points on an elliptic curve over a finite field
• Elliptic curves over $\mathbb{Q}$ can be viewed as curves over $\mathbb{Z}$ (integral models) by clearing denominators in the Weierstrass equation

Modular Forms: Fundamentals

• Modular forms are complex analytic functions $f(z)$ on the upper half-plane $\mathbb{H}$ satisfying $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$ for all $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) \in SL(2,\mathbb{Z})$ and some integer $k$ (the weight)
• Cusp forms are modular forms that vanish at the cusps (points at infinity) of the modular curve $SL(2,\mathbb{Z}) \backslash \mathbb{H}$
• Eisenstein series are examples of non-cuspidal modular forms, given by $G_k(z) = \sum_{(c,d) \neq (0,0)} (cz+d)^{-k}$
• Hecke operators $T_n$ are linear operators acting on the space of modular forms of a given weight and level, and their eigenforms are simultaneous eigenfunctions for all $T_n$
• Modular forms have Fourier expansions $f(z) = \sum_{n=0}^{\infty} a_n e^{2\pi i n z}$, with the coefficients $a_n$ encoding arithmetic information
• Modular curves are algebraic curves associated to congruence subgroups of $SL(2,\mathbb{Z})$, and modular forms can be viewed as sections of line bundles on these curves
• Modular forms have a rich theory of Hecke operators, L-functions, and Galois representations, connecting them to various areas of number theory

Connections Between Elliptic Curves and Modular Forms

• Modularity theorem states that every elliptic curve over $\mathbb{Q}$ is modular, meaning it corresponds to a unique modular form of weight 2 and level equal to the conductor of the curve
• L-function of an elliptic curve $E$ over $\mathbb{Q}$ is the same as the L-function of the corresponding modular form $f_E$
  • Coefficients of $f_E$ are related to the number of points on the reduction of $E$ modulo primes
• Modular parametrization of an elliptic curve is a surjective map from a modular curve to the elliptic curve, induced by the corresponding modular form
• Shimura-Taniyama conjecture (now a theorem) states that the L-function of an elliptic curve over $\mathbb{Q}$ has an analytic continuation and satisfies a functional equation, as predicted by the modularity of the curve
• Eichler-Shimura theory relates the Hasse-Weil L-function of an elliptic curve over $\mathbb{Q}$ to the Mellin transform of the corresponding modular form
• Modular symbols provide a way to compute modular forms associated to elliptic curves, by integrating differential forms on modular curves
• Galois representations associated to elliptic curves and modular forms are compatible under the modularity correspondence, providing a powerful tool for studying their arithmetic properties

Applications in Number Theory

• Fermat's Last Theorem follows from the modularity of semistable elliptic curves, as shown by Wiles and Taylor-Wiles
  • Frey curve associated to a hypothetical solution of Fermat's equation is semistable, but its modularity contradicts Ribet's level-lowering theorem
• Congruent number problem asks which integers are the areas of right triangles with rational side lengths, and is related to the rank of elliptic curves over $\mathbb{Q}$
• Birch and Swinnerton-Dyer conjecture predicts that the rank of an elliptic curve over $\mathbb{Q}$ is equal to the order of vanishing of its L-function at $s=1$, connecting arithmetic and analytic properties
• Elliptic curves are used in the study of Diophantine equations, such as finding integer solutions to equations like $y^2 = x^3 + k$
• Modular forms and elliptic curves are central to the Langlands program, which aims to unify various areas of number theory and representation theory
• Serre's conjecture (now a theorem of Khare-Wintenberger) characterizes the modularity of Galois representations, providing a powerful tool for studying Diophantine equations
• Elliptic curves and modular forms are used in the construction of p-adic L-functions and the study of special values of L-functions, connecting them to Iwasawa theory and other areas of p-adic number theory

Computational Methods and Tools

• Sage, a free and open-source mathematical software system, provides extensive support for working with elliptic curves and modular forms
  • Includes databases of elliptic curves, modular forms, and related objects
• Magma, a commercial computer algebra system, has powerful functionality for computing with elliptic curves and modular forms
• Pari/GP is a widely-used open-source computer algebra system with strong support for number theory, including elliptic curves and modular forms
• Modular symbols algorithms allow for efficient computation of modular forms and their associated L-functions, by working with homology groups of modular curves
• Mestre's algorithm and other methods for finding elliptic curves with a given conductor or other properties
• Schoof's algorithm and its improvements (Schoof-Elkies-Atkin) for counting points on elliptic curves over finite fields
• Modular forms database provides a wealth of examples and data for exploring the connections between elliptic curves and modular forms
  • LMFDB (L-functions and Modular Forms Database) is an extensive online resource

Advanced Topics and Current Research

• Serre's uniformity conjecture on the surjectivity of Galois representations associated to elliptic curves
• Modularity of elliptic curves over more general number fields, beyond $\mathbb{Q}$
• Generalized Fermat equations and the modularity of Galois representations
• Elliptic curves over function fields and their connections to Drinfeld modular forms
• Higher-dimensional analogues of elliptic curves, such as abelian varieties and K3 surfaces, and their connections to Siegel modular forms and automorphic forms
• p-adic and overconvergent modular forms, and their role in the study of p-adic variation of arithmetic objects
• Iwasawa theory of elliptic curves and modular forms, studying their behavior in towers of number fields
• Modularity lifting theorems and their applications to Diophantine equations and Galois representations