discriminant_group.md

Discriminant Groups

CurrentModule = Hecke
DocTestSetup = quote
    using Hecke
  end

Torsion Quadratic Modules

A torsion quadratic module is the quotient $M/N$ of two quadratic integer lattices $N \subseteq M$ in the quadratic space $(V,\Phi)$. It inherits a bilinear form

$$b: M/N \times M/N \to \mathbb{Q} / n \mathbb{Z}$$

as well as a quadratic form

$$q: M/N \to \mathbb{Q} / m \mathbb{Z}.$$

where $n \mathbb{Z} = \Phi(M,N)$ and $m \mathbb{Z} = 2n\mathbb{Z} + \sum_{x \in N} \mathbb{Z} \Phi (x,x)$.

torsion_quadratic_module(M::ZZLat, N::ZZLat)

The underlying Type

TorQuadModule

Most of the functionality mirrors that of AbGrp its elements and homomorphisms. Here we display the part that is specific to elements of torsion quadratic modules.

Attributes

abelian_group(T::TorQuadModule)
cover(T::TorQuadModule)
relations(T::TorQuadModule)
value_module(T::TorQuadModule)
value_module_quadratic_form(T::TorQuadModule)
gram_matrix_bilinear(T::TorQuadModule)
gram_matrix_quadratic(T::TorQuadModule)
modulus_bilinear_form(T::TorQuadModule)
modulus_quadratic_form(T::TorQuadModule)

Elements

quadratic_product(a::TorQuadModuleElem)
inner_product(a::TorQuadModuleElem, b::TorQuadModuleElem)

Lift to the cover

lift(a::TorQuadModuleElem)
representative(::TorQuadModuleElem)

Orthogonal submodules

orthogonal_submodule(T::TorQuadModule, S::TorQuadModule)

Isometry

is_isometric_with_isometry(T::TorQuadModule, U::TorQuadModule)
is_anti_isometric_with_anti_isometry(T::TorQuadModule, U::TorQuadModule)

Primary and elementary modules

is_primary_with_prime(T::TorQuadModule)
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem})
is_elementary_with_prime(T::TorQuadModule)
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem})

Smith normal form

snf(T::TorQuadModule)
is_snf(T::TorQuadModule)

Discriminant Groups

See Nik79 for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice

discriminant_group(::ZZLat)

From a matrix

torsion_quadratic_module(q::QQMatrix)

Rescaling the form

rescale(T::TorQuadModule, k::RingElement)

Invariants

is_degenerate(T::TorQuadModule)
is_semi_regular(T::TorQuadModule)
radical_bilinear(T::TorQuadModule)
radical_quadratic(T::TorQuadModule)
normal_form(T::TorQuadModule; partial=false)

Genus

genus(T::TorQuadModule, signature_pair::Tuple{Int, Int})
brown_invariant(T::TorQuadModule)
is_genus(T::TorQuadModule, signature_pair::Tuple{Int, Int})

Categorical constructions

direct_sum(x::Vector{TorQuadModule})
direct_product(x::Vector{TorQuadModule})
biproduct(x::Vector{TorQuadModule})

Submodules

submodules(::TorQuadModule)
stable_submodules(::TorQuadModule, ::Vector{TorQuadModuleMap})