Number Theory - Elliptic Curves
Mathematics 605C: spring 2011
Lecture: MWF 10:00-10:50, Engineering E104.
Prerequisite: Math 566 or permission of professor.
Questions for group presentations in week 3
Detailed information on homework
Introduction to elliptic curves: cubic equations, the group law, projective space,
Weierstrauss model, j-invariant, complex tori, torsion points.
Weeks 4-5: Endomorphisms, automorphisms, isogenies, complex multiplication.
Weeks 6-8: Modular curves, quotient of upper half plane by congruence subgroups, modular forms.
Weeks 9-10: Rational points, Mordell's theorem, rank.
Weeks 11-12: Galois representations, Tate module, application to abelian extensions of quadratic fields.
Weeks 13-14: Reduction of elliptic curves, supersingular elliptic curves, Zeta functions of elliptic curves, Hecke L-functions.
Week 15: Official proposed version
Version from 2007 including references
The study of number theory originated in ancient civilizations
such as those of China and India and was developed in great depth in Europe
in the 17th and 18th centuries.
Number theory is known for having problems that are easy to state yet which
can only be solved using complicated structures.
For example, it took 300 years to find a complete proof of Fermat's Last Theorem.
Number theory is a subject that's intertwined with group theory, algebraic
geometry, combinatorics, and complex analysis. It's become popular recently
because of its applications to coding theory and cryptography.
Number theory is a vast subject. In this course, we will focus on elliptic curves. Elliptic curves are cubic curves whose points satisfy a group law; they provide the fundamental example of a group variety and are of central importance in number theory and algebraic geometry. The proof of Fermat's Last Theorem relied on modularity results for elliptic curves. They are also important in complex analysis, because a complex torus is an elliptic curve and elliptic functions are functions on complex elliptic curves, and in algebra and representation theory. In addition, they have applications to cryptography and factorization and to Hilbert's 10th problem. In this course, we will study elliptic curves defined over the complex numbers, number fields and finite fields and study some of their applications. Other topics will include zeta functions, Galois representations associated with elliptic curves and modular curves.
The course grade will be based on 20% group presentation, 40% homework and 40% on final project.
The project is an opportunity to learn more about a topic in number theory that interests you or will be relevant for your future graduate work. It gives us a chance to hear about important ideas which we will not have time to cover in class. It is also a good opportunity to develop more skill at writing and speaking on mathematics.
Possible topics for final presentations: Pairings and applications to elliptic curve cryptography
Integral points and Nagell-Lutz Theorem
Birch and Swingerton-Dyer Conjecture
Applications to Hilbert's 10th problem
Fermat's Last Theorem
Complex abelian varieties
Help: Help is always available if you have trouble with homework or lecture material. If your classmates can't answer your question, come ask me! Office hours will be (TBA) or are available by appointment.