Differences between revisions 4 and 6 (spanning 2 versions)
 ⇤ ← Revision 4 as of 2009-09-08 21:16:49 → Size: 1200 Editor: was Comment: ← Revision 6 as of 2009-09-08 22:10:04 → ⇥ Size: 3054 Editor: was Comment: Deletions are marked like this. Additions are marked like this. Line 7: Line 7: * URL: http://wstein.org/Tables/real_tamagawa/ * URL: http://wstein.org/Tables/real_tamagawa/ and http://wstein.org/Tables/compgrp/. The second page has much more extensive data and a conjecture. Line 35: Line 35: * URL: http://wstein.org/Tables/cuspgroup/ * URL: http://wstein.org/Tables/cuspgroup/  (the displayed formula is backwards at the top) Line 54: Line 54: == Discriminants of Hecke Algebra ==Computation of discriminants of various Hecke algebras. URL:    * http://wstein.org/Tables/dischecke.htmlAmazingly, 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 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 (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.

# Specific Tables

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

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.

## Cuspidal Subgroup

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

```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.

• URL:

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 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 (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.

days17/projects/presagedays/discussion (last edited 2010-07-12 07:39:51 by was)