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