MATH 7290 - Spring 2013

Elliptic Curves and Modular Forms

This is a first course in elliptic curves and modular forms, which are at the heart of modern number theory.  See the Syllabus and detailed Lecture Schedule for more details.

Course Information

Scheduled Time
Lectures TR 1:30
Lockett 135
Office Hours W 2:00 - 3:00
Lockett 228

Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edition.

Problem Sets

Lecture Schedule

Lecture Topics / Reading / Handouts
Tues., Jan. 15
Motivating Examples: Rational points on curves, Diophantine equations, Sums of squares
Thurs., Jan. 17
Congruent numbers (Koblitz 1.1); A related cubic curve (Koblitz 1.2)
Tues., Jan. 22
Definition of elliptic curves, Basic properties of algebraic and projective varieties (Koblitz 1.3)
Thurs., Jan. 24
Visualizing projective space, smooth varieties, Addition law for elliptic curves (Koblitz 1.3)
Tues., Jan. 29
Addition law for elliptic curves, closure of rational points, points of finite order and
Mazur's theorem, Mordell-Weil theorem (Koblitz 1.7)
Thurs., Jan. 31  
Algebraic varieties, birational maps, fields of functions
Tues., Feb. 5
Complex projective space, conics; Real periodic functions
Thurs., Feb. 7
Laurent series for Cotangent function, even zeta values; Complex lattices,
Weierstrass elliptic P-function (Koblitz 1.4)
Thurs., Feb. 14
Complex analysis and elliptic functions, zeros and poles of Weierstrass P-function (Koblitz 1.4); Field of elliptic functions and P-function (Koblitz 1.5); Parametrization of elliptic curves (Koblitz 1.6)
Tues., Feb. 19
Eisenstein series and P-function (Koblitz 1.6); Complex structure of addition law (Koblitz 1.7)
Thurs., Feb. 21
Points of finite order on elliptic curves, Galois theory of coordinates (Koblitz 1.8); Elliptic curves over finite fields (Koblitz 1.9)
Tues., Feb. 26
Torsion group of congruent number curves (Koblitz 1.8, 1.9); Definition of Zeta-functions (Koblitz 2.1)
Thurs., Feb. 28
Calculating zeta-functions via character sums, Gauss and Jacobi sums (Koblitz 2.2; survey)
Tues., Mar. 5
Properties of Gauss and Jacobi sums, finite field extensions and Hasse-Davenport relations (Koblitz 2.2)
Thurs., Mar. 7
Hasse-Weil L-function of elliptic curve (Koblitz 2.5); Analytic continuation and function equation for Riemann zeta function, Mellin transform, Poisson summation (Koblitz 2.4; survey)
Tues., Mar. 12
Elliptic curve cryptography (survey); Inversion formula for theta function (Koblitz 2.4)
Thurs., Mar. 14
Comments on Hasse-Weil L-function of elliptic curves (Koblitz 2.5); Real points on elliptic curves; Defintion of modular forms (Koblitz 3.2); First properties of linear fractional transformations (Koblitz 3.1)
Tues., Mar. 19
Fundamental domain for the modular group; Generators of the modular group (Koblitz 3.1); Eisenstein series (Koblitz 3.2)
Thurs., Mar. 21
Delta function, relation to discriminant of elliptic curves; Valence formula; Structure of graded ring of modular forms (Koblitz 3.2)
Tues., Mar. 26
Parametrization of modular functions with j-function (Koblitz 3.2)
Thurs., Mar. 28
Modular transformations for Eisenstein series of weight 2 (Koblitz 3.2)
Tues., Apr. 9
Examples of parametrization of modular functions with j-function (MAPLE file); Dedekind's eta-function, Pentagonal Number Theorem, Jacobi Triple Product (overview); Product formula for Delta function (Koblitz 3.2)
Thurs., Apr. 11 Hecke operators on lattic functions; Basic properties and commutativity (Koblitz 3.5)
Tues., Apr. 16
U_m and V_m operators; Formal Dirichlet series of Hecke operators; Action of Hecke operators on coefficients of modular forms (Koblitz 3.5)
Thurs., Apr. 18
Examples of action of Hecke operators; Eigenforms in one-dimensional spaces; Diagonalizing cusp forms of weight 24 (MAPLE file); Normalized Hecke eigenforms; Vector spaces and Hermitian operators; Definition of Petersson scalar product (Koblitz 3.5)
Tues., Apr. 23
Properties of Petersson scalar product, Hecke operators as hermitian operators (Koblitz 3.5); Definition of congruence subgroups (Koblitz 3.1)
Thurs., Apr. 25
Fundamental domains for congruence subgroups, cusps, Modular forms on congruence subgroups, U, V and Hecke operators (Koblitz 3.3)
Tues., Apr. 30
One-dimensional spaces of cusp forms, Eisenstein series on congruence subgroups, twists of modular forms by Dirichlet characters (Koblitz 3.3); Modular forms of half-integral weight (Koblitz 4.1); Examples of modular elliptic curves (MAPLE file)
Thurs., May 2
Applications of modular forms to partitions; asymptotic formulas, linear congruences (MAPLE file)

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