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

== Status report 28.06.2010 == See http://wiki.sagemath.org/days22/greenberg/june282010

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

(24.06.2010 update)

Attached are some text files giving some data about the minimal conductor < 10,000 for an elliptic curve realizing a mod 2, 3, or 5 representation attached to a newform of level < 150. -Sam

norm2cond10000 norm3cond10000 norm5cond10000

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

(24.06.2010 update)

Level 29 gives an example. Using the Hasse bound we see that a2 is -2,-1,0,1,2, so a2 mod 7 is 0,1,2,5,6. Thus one of the level 29 forms doesn't come from an elliptic curve.

D. Prime powers for small primes

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

(24.06.2010 update)

Gagan reports that the Galois representations associated to 121A and 121C are surjective mod 2 but not mod 4.

There is only one conjugacy class of subgroups of GL(2,Z/9) which surjects onto (Z/9Z)* under the determinant map and reduces onto GL(2,Z/3). It is of size 144=3*#GL(2,Z/3). So Elkies' analysis of "3 not 9" pins down the image of Galois completely.