Please can we also have real-valued bilinear forms? Mordell-Weil
lattices (from elliptic curves over number fields) will require
non-rational values.
John
2008/4/28 Robert Miller <[EMAIL PROTECTED]>:
>
> SEP
>
> Implement Lattices over ZZ, with pairings into QQ or ZZ
>
> 0. (Maybe) Implement a FreeModule_ZZ_quotient class. This would also
> allow for constructing abelian groups in the sort of canonical way
> (something people have been asking for...)
>
> 1. Implement a LatticeModule class, which will inherit from
> FreeModule_generic_pid: instances of LatticeModule will inherit an
> underlying free ZZ module and make use of the optional
> inner_product_matrix property.
>
> This shouldn't just be a free ZZ module with inner product matrix,
> since we want specific functions for computing the dual lattice, etc.
> which are more appropriate in a Lattice class.
> a. Attributes will include
> - is_euclidean (whether the inner product matrix is symmetric,
> rather than skew-symmetric)
> - is_integral (whether the image of the pairing is in ZZ or QQ)
> - discriminant (the determinant of the matrix [<a_i,a_j>], where
> {a_i} is a basis for the module). A lattice is nondegenerate if its
> discriminant is nonvanishing.
> b. Euclidean lattices also have the attributes:
> - signature
> - even/odd (whether <a,a> \in 2 ZZ for all a)
> c. Use L.<a,b> for the pairing induced on module elements by the
> inner product matrix.
>
> 2. Implement a SubLatticeModule class, which will inherit from
> FreeModule_submodule_with_basis_pid and from Lattice, but override
> L.<a,b> for the inner product.
> a. Function is_primitive (a sublattice M of a lattice L is
> primitive if L/M is a free ZZ-module)
> b. Functions to get parent lattice and sublattice as LatticeModule
> objects.
>
> 3. Implement a LatticeQuotient class (for now, just full sublattices,
> i.e., finite quotients).
> -- Inherit from FreeModule_ZZ_quotient?
> -- Inherit from AbelianGroup?
> -- Inherit from nothing?
> ( The question here is what the underlying structure for a
> LatticeQuotient should actually be. The important thing is how will
> someone want to access elements of a LatticeQuotient? )
> a. Attributes will include a quadratic_form_matrix with entries
> defined over QQ/ZZ or QQ/2ZZ
>
> 4. Create a dual_lattice function for integral euclidean lattices,
> with optional "embedding" argument
>
> 5. Implement a dual_quotient function for integral euclidean lattices
> which returns a LatticeQuotient.
>
> 6. Implement isomorphism tests for indefinite integral euclidean
> lattices.
>
> -- Robert Miller, Andrey Novoseltsev, Ursula Whitcher
>
> >
>
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