Specific Tables

Component Groups of J0(N)(R) and J1(N)(R)

• URL: http://wstein.org/Tables/real_tamagawa/ and http://wstein.org/Tables/compgrp/. The second page has much more extensive data and a conjecture.

• New Code:

This function computes the J_0(N) real component groups.

def f(N):
    M = ModularSymbols(N).cuspidal_subspace()
    d = M.dimension()//2
    S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list()) - 1
    return 2^(S.nullity()-d)

For J_1(N) it is:

def f(N):
    M = ModularSymbols(Gamma1(N)).cuspidal_subspace()
    d = M.dimension()//2
    S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list()) - 1
    return 2^(S.nullity()-d)

Future extension: one could replace Gamma1(N) by GammaH(N,...). One could also do the new subspace.

And note Frank's conjecture:

Conjecture (Frank Calegari):

Let m = #odd prime factors of N +  {1, if N = 0 mod 8
                                   {0, otherwise.
Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m - 1.

the above conjecture is wrong, but the following matches our data (up to level N<=2723):

Conjecture (Boothby-Stein):
Let m = #odd prime factors of N -  {1, if N != 0 mod 8
                                   {0, otherwise.

Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m - 1, unless N=1,2,4, in which case the component is

Soroosh -- the prime level case is known. See Calegari <insert link> which *just* cites Agashe and Merel (http://wstein.org/home/wstein/days/17/Merel_Laccouplement_de_Weil_entre_le_sous-group.pdf -- page 12).

Cuspidal Subgroup

Computing the structure of the cuspidal subgroup of J0(N) and J1(N) (say).

• URL: http://wstein.org/Tables/cuspgroup/ (the displayed formula is backwards at the top)

• New Sage code:

def cuspidal_subgroup_J0(N):
    J = J0(N)
    I = J.cuspidal_subgroup().invariants()
    # maybe pickle J
    return I

def cuspidal_subgroup_J1(N):
    J = J1(N)
    I = J.cuspidal_subgroup().invariants()
    # maybe pickle J
    return I

BUT WAIT -- isn't there an a priori formula for this structure/order? Yes -- Ligozat, but not really -- that gives only rational cuspidal subgroup, and might be just as hard.

Anyway, I'm computing a few of these here, as a test of the modular symbols code, etc., since this is easy:

http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J0N/

http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J1N/

Discriminants of Hecke Algebra

Computation of discriminants of various Hecke algebras.

• URLs:

  • http://wstein.org/Tables/dischecke.html

  • http://trac.sagemath.org/sage_trac/ticket/6635

Amazingly, it seems that there is "discriminants of Hecke algebras" implementation in Sage! Here is a straightforward algorithm:

1. The input is the level N.

2. Chose a random vector v in the space M of cuspidal modular symbols of level N.

3. Compute the sturm bound B.

4. Compute the products T_1(v), ..., T_B(v), and find a basis b_i for the ZZ-module they span.

5. Find Hecke operators S_1, ..., S_n such that S_i(v) = b_i. (This is linear algebra -- inverting a matrix and a matrix multiply.)

6. Compute the determinant det ( Trace(S_i * S_j) ). That is the discriminant. This also gives a basis for the Hecke algebra, which is very useful for lots of things.

Note: See http://trac.sagemath.org/sage_trac/ticket/6768 for very slow code for computing a basis for the Hecke algebra.

Here is a more complicated algorithm, but it might suck because of hidden denseness!

1. The input is the level N.

2. If N is divisible by a prime p^3 and X_0(N/p^3) has positive genus, then the discriminant is 0, as one can see by taking images of forms of level N/p^3.

3. I think the above is an if and only if condition for when the discriminant is 0. See I think Coleman-Voloch.

4. The actual algorithm now.

   1. Find a random Hecke operator t such that the charpoly of t has nonzero discriminant.

   2. Choose a random vector v in the space of cuspidal modular symbols.

   3. Let B be the Sturm bound.

   4. Compute the images T_n(v) for n up to the Sturm Bound.