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| 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 is complete in the sense that the Galois images is pinned down. | 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. |
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
(24.06.2010 update)
Here are a couple of examples of mod 5 representations for which the elliptic curve (which must exist of course!) is rather large. ("up to primes below 10000" refers to how high I'm comparing the coefficients, rather than worrying about the sturm bound.) If anyone has the optional conductors up to 130000 database installed and wants to try running my code to find the curves, let me know and I'll send you my program.
Testing curves of conductor < 10000 up to primes below 10000 Testing a newform of level 31 with coefficient field Number Field in a with defining polynomial x^2 - x - 1 with prime Fractional ideal (-2*a + 1) of norm 5 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (-2*a + 1) No elliptic curve matched the form!
Testing a newform of level 41 with coefficient field Number Field in a with defining polynomial x3 + x2 - 5*x - 1 with prime Fractional ideal (1/2*a^2 + a - 5/2) of norm 5 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (1/2*a^2 + a - 5/2) No elliptic curve matched the form!
Here is some more interesting data: for newforms of levels [29..100] with non-rational hecke field with a prime of norm 2, the following are (I think) the only examples where the level seems to be nonoptimal (i.e. serre conductor is strictly smaller than the level, i.e. I found a curve of conductor not divisible by the level).
Testing a newform of level 63 with coefficient field Number Field in a with defining polynomial x^2 - 3 with prime Fractional ideal (a - 1) of norm 2 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (a - 1) Curve 14a1 of conductor 2 * 7 succeeded!
Testing a newform of level 88 with coefficient field Number Field in a with defining polynomial x^2 - x - 4 with prime Fractional ideal (a + 1) of norm 2 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (a + 1) Curve 11a1 of conductor 11 succeeded!
Testing a newform of level 93 with coefficient field Number Field in a with defining polynomial x3 - 4*x + 1 with prime Fractional ideal (a2 - 3) of norm 2 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (a^2 - 3) Curve 2325b1 of conductor 3 * 5^2 * 31 succeeded!
Testing a newform of level 98 with coefficient field Number Field in a with defining polynomial x^2 - 2*x - 7 with prime Fractional ideal (1/2*a - 1/2) of norm 2 Attempting to reduce coefficients Reduced the coeffs mod Fractional ideal (1/2*a - 1/2) Curve 49a1 of conductor 7^2 succeeded!
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.