Course Information |
Scheduled
Time |
Room |
| Lectures | TR 1:30 |
Lockett
135 |
| Office Hours | W 2:00 -
3:00 |
Lockett
228 |
| Textbook |
Neal
Koblitz, Introduction to Elliptic Curves and Modular Forms,
2nd edition. |
| Date |
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) |