Topics in Number Theory: Rational points on Algebraic Curves
Knowledge of standard first year graduate algebra and complex analysis
Arithmetic of Elliptic Curves by J.H. Silverman
The study of rational solutions to Diophantine equations of the form F(x,y)=0 for a polynomial F with coefficients in a field K arise often in number theory and correspond geometrically to points on an algebraic curve. The cases when F(x,y)=y^2-f(x) (where f(x) is a cubic polynomial with distinct roots) are called elliptic curves and are of special interest since the set of points with coordinates in a field L can be given a group law. We will study this group for L the complex numbers, finite fields, local fields and number fields. This structure allows an array of technical tools such as representation theory, Galois cohomology, group schemes, and Selmer groups to be utilized to analyze Diophantine problems.
These tools may be used to analyze specific number theoretic problems, such as determining all integers which are the areas of right triangles with integer sides.
These lead to open problems and relations with L-series and modular forms.