**Number Theory - Elliptic Curves **

Mathematics 605C: spring 2018

web page: www.math.colostate.edu/~pries;

office: Weber 205A. office hours (tentative) Wed 10-11, Thurs 1-2, or by appointment.

** Lecture:** MWF 9:00-9:50, Weber 202.

**Prerequisite:**
Math 566 or permission of professor.

**Homework:**
**
Detailed information on homework**

**Tentative Syllabus:**

** Weeks 1-3:**
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: Final presentations **

Official proposed version

**Course description:**
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.

** Textbooks:**
This textbook is free and legally available online:
Milne, Elliptic Curves

We will also use chapters of the following books, which are freely available to CSU students through SpringerLink:

step 1: go to the CSU library page CSU library page
or (on-campus only?) link.springer.com .

step 2: in the almost everything box, type in keywords
(e.g. Everest and Ward and Introduction and Number) and click the search icon.

step 3: click on the item, look for full text available, and download.

primary: Silverman, The Arithmetic of Elliptic Curves

primary: Cohen, Number Theory, Volume 1, part II Diophantine Equations

Ok, this is too many books for one semester, but they may be useful for projects.

secondary: Koblitz, Introduction to Elliptic Curves and Modular Forms

secondary: Silverman, Advanced Topics in Elliptic Curves

secondary (easier): Silverman and Tate, Rational Points on Elliptic Curves

secondary (easier) : Hindry, Arithmetics, Chapter 5

secondary (easier) : Ireland and Rosen, A Classical Introduction to Modern Number Theory, second edition, Chapters 17-20

secondary (harder): Hida, Elliptic curves and arithmetic invariants, Chapter 2, Shimura varieties

secondary (harder): Cornell, Silverman, Stevens, Modular forms and Fermat's Last Theorem

**Grading:**
The course grade will be based on 20% short presentations, 40% homework and 40% on final project.

** 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.

** Some ideas for topics for final projects:**
(more later); pairings; heights; elliptic curve cryptography;
integral points and the Nagell-Lutz Theorem;
the 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 are Wed 10-11, Thurs 1-2, or by appointment.