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|2. Chose a random vector $v$ in the space $M$ of modular symbols of level $N$.||2. Chose a random vector $v$ in the space $M$ of cuspidal modular symbols of level $N$.|
Component Groups of J0(N)(R) and J1(N)(R)
- 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.
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 = C.cuspidal_subgroup().invariants() # maybe pickle J return I
def cuspidal_subgroup_J0(N): J = J1(N) I = C.cuspidal_subgroup().invariants() # maybe pickle J return I
Discriminants of Hecke Algebra
Computation of discriminants of various Hecke algebras.
Amazingly, it seems that there is "discriminants of Hecke algebras" implementation in Sage! Here is a straightforward algorithm:
The input is the level N.
Chose a random vector v in the space M of cuspidal modular symbols of level N.
Compute the sturm bound B.
Compute the products T_1(v), ..., T_B(v), and find a basis b_i for the ZZ-module they span.
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.)
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!
The input is the level N.
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.
I think the above is an if and only if condition for when the discriminant is 0. See I think Coleman-Voloch.
- The actual algorithm now.
Find a random Hecke operator t such that the charpoly of t has nonzero discriminant.
Choose a random vector v in the space of cuspidal modular symbols.
Let B be the Sturm bound.
Compute the images T_n(v) for n up to the Sturm Bound.