Modular Abelian Varieties

Code at http://trac.sagemath.org/sage_trac/ticket/2544

Todo on Tuesday, March 18

(craig) 3: Hom(A,B) for A, B simple
DONE (craig) 2: disc(End(A))
DONE (craig) 2: degeneracy maps
DONE (craig) 1: fix hecke operators
(robert) 3: Hom(A,B) in general
DONE (robert) 1: create newforms and abvars from label, where label is 389aG1, 389aGH[2,3], 389b, 389c, etc.
DONE (robert) 1: complementary isogeny (invert matrix and clear denom)
(william) 2: cokernels of morphisms
(william) 3: dual isogeny when A,B both maximal
(william) 2: quotient by abelian subvariety
(william) 2: dual of A when A is maximal
DONE (william) 2: kernels of morphisms
DONE (william) 2: projection (only when 'maximal')
(william) images of abelian varieties and groups under morphisms.
(group) 2: norm equations (clean up patch)
(group) 5: minimal isogeny degree

Todo on March 16

DONE (craig) Newforms issue:

   sage: f = CuspForms(37).newforms('a')[0]
   sage: f.coefficients(10)
   ------------------------------------
   <type 'exceptions.TypeError'>             Traceback (most recent call last)
   sage: f.coefficients([2..10])
   ------------------------------------
   <type 'exceptions.AttributeError'>        Traceback (most recent call last)
   ...
   <type 'exceptions.AttributeError'>: 'Newform' object has no attribute '_compute'

DONE (william) This is completely wrong (the modabvar function on modular symbols assumes it's ambient!):

   sage: m = ModularSymbols(37)[1]
   sage: m.modular_abelian_variety()
   Jacobian of the modular curve associated to the congruence subgroup Gamma0(37)

DONE (william) Move functions out of abvar_modsym_factor into abvar and delete that file.
DONE (william) Torsion subgroups:
- (already done) Refactor base class
- (done) Get implementation to work with defining data being (lattice, abvar); compute generators.
- (done) Quotients by finite subgroup
DONE (craig) Implement f.number() for f a newform.
DONE-ish (craig) Compute the Hecke algebra image in End(A) and find a good clean way to represent for Hecke stable. New object that is a subring of End(A). Have methods like R.index_in(S).
DONE (craig) abelian varieties should cache their ambient modular abelian variety

Todo on Monday, March 17

(craig) Compute End(A):
- for simple A_f (DONE)
- in general.
- disc of it.
- ideals and annihilators
- order in a ring of integers (for A simple)
- given a ring R in EndAlg(A) where every element has integral charpoly, find an isogenous abelian variety with End(A) = R.
- explicit isomorphism between HeckeAlgebra sitting in End(A) and a commutative ring
- base extension of End(A)
(craig) Degeneracy maps
(william) Decomposition:
- three types:
  - ungrouped as simple abvars (default)
  - groups abvars
  - over End(A)
  - deprecate hecke_decomposition
- (DONE) label function
- create from label
(william/craig) Morphisms:
- Kernels
- Cokernels
(william/craig) Isogenies:
- Complementary -- invert matrix, clear denom.
- Dual
(william) Intersection pairing
(?) Poincare Reducibility:
- projection
- quotients by abelian subvariety
(craig/william) Minimal isogeny degree for A, B simple.
(craig/william) Create a small mock database

Todo on Wednesday, March 19

Write doctests, etc., for everything above.
Optimize everything