# 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/

## Final presentation, Friday 02.07.2010

See http://wiki.sagemath.org/days22/greenberg/presentation

## Status report, Monday 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.

## Status report, Monday 30.06.2010

See http://wiki.sagemath.org/GaganSekhon

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.

## Misc

A program for finding certain integral modular symbols that Matt G. wanted: http://nt.sagenb.org/home/pub/13/

All of the code Ben Linowitz produced during the conference: Sage worksheet (sws)

All the code and data that William Stein computed when making a big table of newforms: http://sage.math.washington.edu/home/wstein/db/modsym-2010

William's improvements to reduction mod primes code in sage: http://trac.sagemath.org/sage_trac/ticket/9400