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,
MordellWeil 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 Pfunction (Koblitz 1.4) 
Thurs., Feb. 14 
Complex
analysis and elliptic functions, zeros and poles of
Weierstrass Pfunction (Koblitz 1.4); Field of elliptic
functions and Pfunction (Koblitz 1.5); Parametrization of
elliptic curves (Koblitz 1.6) 
Tues., Feb. 19 
Eisenstein
series and Pfunction (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 Zetafunctions (Koblitz 2.1) 
Thurs., Feb. 28 
Calculating
zetafunctions via character sums, Gauss and Jacobi sums
(Koblitz 2.2; survey) 
Tues., Mar. 5 
Properties
of Gauss and Jacobi sums, finite field extensions and
HasseDavenport relations (Koblitz 2.2) 
Thurs., Mar. 7 
HasseWeil
Lfunction 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 HasseWeil Lfunction 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 jfunction (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 jfunction (MAPLE file);
Dedekind's etafunction, 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 onedimensional
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 
Onedimensional
spaces of cusp forms, Eisenstein series on congruence
subgroups, twists of modular forms by Dirichlet characters
(Koblitz 3.3); Modular forms of halfintegral 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) 