Status Report 29-11-2007

Modelling finite quadratic modules as classes

• A model for a finite quadratic module comprises three sorts of objects:
  1. The quadratic module
  2. Its elements
  3. Its subgroups

The quadratic module

• A quadratic module is an abelian group $M$ and a quadratic form $Q:M \rightarrow {\mathbb Q}/{\mathbb Z}$
• An abelian group is given by generators $g_1,\dots,g_n$ and relations.
• Relations are given by a matrix.
• $Q$ is given by a Gram matrix with respect to the generators.
• Conclusion: class quad_module with __init__( self, matrix_of_relations, Gram_matrix)

• An element of a quadratic module is a linear combination of generators.
• Conclusion: class quad_module_element with __init__( self, quad_module, coefficients_of_lin_comb)
Additionally: quad_module.gens[i], __add__ and relatives,
• and Coercion: If A is an instance of class quad_module then A(vector) returns the linear combination of the genarators where vector denotes the coefficients.

Subgroups of a quadratic module

• A subroup of a quadratic module is a list of generators of this subgroup.
• Conclusion: class quad_module_subgroup with __init( self, list_of_gens)
• Additionally: quad_module.subgroup( list_of_gens), __add__, dual
• and Coercion: If A is an instance of class quad_module then A(matrix) returns the subgroup generated by the elements of A whose coordinates are given by the columns of matrix.

Status

All three classes are implemented. The main methods are functional.

Implementation notes

When a quad_module is created the matrix of relations $M$ is replaced by the matrix $D$ of its elementary divisors: $$ D = RMS \qquad (R,S \in \text{GL}(n,{\mathbb Z}) $$ The (given) generators are supposed to satisfy the relations given by $D$. Accordingly, the given Gram matrix $G$ is replaced by $G[R^{-1}]$.

Subgroups of the underlying quadratic module ${\mathbb Z}^n/M{\mathbb Z}^n$ are in one to one correspondence to lattices $M{\mathbb Z}^n \subseteq L \subseteq {\mathbb Z}^n$ (here $n$ denotes the size of $M$). We can write $L=N{\mathbb Z}^n$ fo a suitable regular $n\times n$ matrix $N$, which is unique up to right multiplication by unimodular matrices. Hence we may assume that $N$ is in (say, lower) Hermite normal form (HNF), which makes it unique. Note that $N^{-1}M$ is integral.

Thus subgroups are represented internally by (lower) HNF matrices left dividing $M$. At initialisation time this unique matrix is computed from the given list of generators for the subgroup to be created.

Using SAGE internal functions and a reasonable design for our classe the matrix manipulations are reduced to a minimum. This ensures a as much as possible separation of logical structure from actual data representation for the sake of easy extensibility and maintenance of our code.

Open Problems

1. How to use pydoc or Is there a SAGE doc tool?
2. __radd__ does not work correctly
3. Propagation of generator names
4. User defined generator names
5. Implementation of periodic funtions

Todo

1. Jordan decomposition of a $p$-module
2. qfmodule( number_field_element)
3. Propagate generator names
4. Implement class chi_invariant and, say, the method quad_module.chi_invariant()