2011 Mathematics REU at UW

June 20 - August 12, 2011

• Organizers: Jim Morrow (Inverse Problems) and William Stein (Elliptic Curves / Sage)

• Mailing list: http://groups.google.com/group/uwmathreu

• Application Form

• Inverse problems page

Elliptic Curves

• Participants: Radoslav Kirov, William Stein, Ben LeVeque, Andrew Ohana, Paul Sharaba, Ari Klages-Mundt, Ashwath Rabindranath, Simon Spicer, Aly Deines, Jon Bober

• Specific Problem List

The elliptic curves REU will consist of projects involving elliptic curves, modular forms, and databases, and make extensive use of Sage. It will be organized by William Stein, and the number theory postdoc Jon Bober and grad students Alyson Deines and Simon Spicer will also likely help out. There will be about 6 students.

REU Background Reading List

Don't worry if you can't read or understand everything here. Just read what you can, and generally skim around. There will be lots of talks during the REU to get you up to speed.

• Stein's Algebraic Number Theory, a Computational Approach

• Birch's article that introduces the Birch and Swinnerton-Dyer Conjecture

• Wiles's article on the Birch and Swinnerton-Dyer Conjecture which can be found on the Clay Mathematics Institute website

• Tate's beautiful 1974 article on elliptic curves and Gross's update to it.

• Stein's book (mainly Chapter 1)

• Cremona's book (mainly Chapters 1 and 3)

• Dembele-Voight: Explicit methods for Hilbert modular forms

• Open Source Software: the Sage, Python, and Cython tutorials.

• Silverman's book The Arithmetic of Elliptic Curves is very generally relevant to much of what we will do.

Topics may include

• Main topic: Creating tables of elliptic curves over the field \mathbf{Q}({\rm sqrt}{5}), e.g., as described in this talk, along with extensive data related to the Birch and Swinnerton-Dyer conjecture about each curve, and investigating related conjectures. This will involve both theoretical work and coding projects.

• Computing additional data about elliptic curves over \mathbf{Q}, thus extending the data in http://db.modform.org/

• Computing tables of modular forms (see my book).