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= Matthew Greenberg (University of Calgary): Mod p representations associated to elliptic curves= = Noam Elkies (Harvard) and Matthew Greenberg (University of Calgary): Mod p representations associated to elliptic curves =
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[[attachment:project.pdf|Project description [pdf]]] [[attachment:project.pdf|Project description (pdf)]]

[[attachment:activitysheet.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. S4-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

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. S4-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

days22/greenberg (last edited 2010-07-02 22:13:14 by was)