| First meeting: | Friday, August 29, 11:00AM-12:20PM Goldsmith Hall, Room 226 | |
| Instructor: | Carl Wang Erickson | Office: Goldsmith 206 |
| cwe@brandeis.edu | Office Hours: Tues/Fri, 1pm-2pm |
The study of elliptic curves allows for an especially wide variety of techniques to be applied toward interesting results and computations -- you can see many of them named in the outline in the syllabus. Our overall goal is to study the rational points on elliptic curves. We will start out with algebro-geometric background necessary to define elliptic curves and maps between them. We will then proceed through the main number-theoretic content of the course: the points on elliptic curves defined over various fields -- finite fields, p-adic local fields, and number fields. One of the main goals is to prove the Mordell-Weil theorem, showing that the points of an elliptic curve over a number field have a nice structure. Then, we will do some calculations with rank. Throughout these studies, explicit examples and computations are readily available, and we will often explore them. With what time is left, we may explore further topics, such as modular forms and the modularity of elliptic curves; or the Birch and Swinnerton-Dyer conjecture.
Each student is asked to give a short talk on a topic of their choice. Details and options for topics will be discussed individually. Initial ideals are given in the syllabus.
Further information on background and topics are available in the syllabus.
Full course information and a more detailed course description are available in the syllabus.