# Spring 2013 MATH 191 001 SEM

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 SEM | MWF 12-1P | 5 EVANS | COLEMAN, R F | 54283 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | NONE | Limit:16 Enrolled:1 Waitlist:0 Avail Seats:15 [on 06/26/13] |

**Prerequisites:** Math 113

**Syllabus:** The topic for this Math 191 is an introduction to elliptic curves and modular forms

Course Description: Elliptic curves and modular forms are seemingly disparate objects which are (each on their own) wildly intricate, theoretically rich and of the utmost importance to modern number theory. In fact, the two types of objects share an intimate and non-obvious relationship - a relationship deep enough to imply a result whose proof eluded the world's top mathematicians for more than 350 years.

We will give an introduction to curves, coordinate rings, projective and affine space, elliptic curves, their group law, elliptic curves over various fields, maps between elliptic curves, differentials. Divisors, Jacobians, elementary moduli of elliptic curves over the complex numbers, and the Mordell-Weil theorem.

We will also talk about lattices, congruence subgroups, modular forms, double coset and Hecke operators, Eisenstein series, Fourier expansions, the Peterson product, and the statement of a first version of the modularity theorem.

**Office:** 901 Evans

**Office Hours:** Friday 3-4

**Required Text:** *Elliptic Curves* by J.S. Milne (See http://www.jmilne.org/math/Books/ectext0.pdf) and *Rational Points on Elliptic Curves* by Joseph Silverman and John Tate

**Recommended Reading: **Silverman - The Arithmetic of Elliptic Curves

**Grading:** To Be Announced

**Homework:** To Be Announced

**Course Webpage:** math.berkeley.edu/~coleman/Courses/Sp13/ecmf.html