Noam Elkies (Harvard) and Matthew Greenberg (University of Calgary): Mod p representations associated to elliptic curves

Project description (pdf)

Activity sheet (pdf)

Background reading:

Silverman, "The arithmetic of elliptic curves", Chapters 3 and 7

Diamond and Shurman, "A first course in modular forms, Chapter 9

Neukirch, "Algebraic number theory", Chapter 2, Section 10 and Chapter 5, Section 6

Ribet and Stein, "Lectures on Serre's conjecture", Chapter 1, see http://wstein.org/papers/serre/

Projects

A. Find the elliptic curve that modular mod-p representations come from, for p < 7

People: William Stein, Mike Lipnowski, Sam Lichtenstein, Ben Linowitz, Laura Peskin, David Ai, Rodney Keaton, M. Tip, Brandon Levin

B. S_4-extensions: find the curves

People: Brandon Levin, Mike Lipnowski, Gagan Sekhon, Noam Elkies, Jon Cass, David Ai

C. Mod-7 galreps from abvars of prime level not arising from elliptic curves

People: Laura Peskin, M. Tip, Arijit, Rebecca, Mike D, Noam

D. Prime powers for small primes

People: Ben Linowitz, Sam Lichtenstein, Gagan, Chris Wuthrich, Barinder, Hatice