Modular Abelian Varieties

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

Todo on Tuesday, March 18

• (craig) 2: disc(End(A))
• (craig) 2: degeneracy maps
• (craig) 3: Hom(A,B) for A, B simple
• (craig) 1: fix hecke operators
• (robert) 1: create newforms and abvars from label, where label is 389aG1, 389aGH[2,3], 389b, 389c, etc.
• (robert) 3: Hom(A,B) in general
• (robert) 1: complementary isogeny (invert matrix and clear denom)
• (william) 2: kernels of morphisms
• (william) 2: cokernels of morphisms
• (william) 3: dual isogeny when A,B both maximal
• (william) 2: quotient by abelian subvariety
• (william) 2: projection (only when 'maximal')
• (william) 2: dual of A when A is maximal
• (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 Af (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