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Genera of Integer Lattices

CurrentModule = Hecke
DocTestSetup = quote
    using Hecke
  end

Two $\mathbb{Z}$-lattices $M$ and $N$ are said to be in the same genus if their completions $M \otimes \mathbb{Z}_p$ and $N \otimes \mathbb{Z}_p$ are isometric for all prime numbers $p$ as well as $M \otimes \mathbb{R} \cong N\otimes \mathbb{R}$.

The genus of a $\mathbb{Z}$-lattice is encoded in its Conway-Sloane genus symbol. The genus symbol itself is a collection of its local genus symbols. See CS99 Chapter 15 for the definitions. Note that genera for non-integral lattices are supported.

The class ZZGenus supports genera of $\mathbb{Z}$-lattices.

ZZGenus

Creation of Genera

From an integral Lattice

genus(::ZZLat)

From a gram matrix

genus(A::MatElem)

Enumeration of genus symbols

integer_genera(sig_pair::Tuple{Int,Int}, determinant::Union{Int,ZZRingElem})

From other genus symbols

direct_sum(G1::ZZGenus, G2::ZZGenus)

Attributes of the genus

dim(G::ZZGenus)
rank(G::ZZGenus)
signature(G::ZZGenus)
det(G::ZZGenus)
iseven(G::ZZGenus)
is_definite(G::ZZGenus)
level(G::ZZGenus)
scale(G::ZZGenus)
norm(G::ZZGenus)
primes(G::ZZGenus)
is_integral(G::ZZGenus)

Discriminant group

discriminant_group(::ZZGenus)

Primary genera

is_primary_with_prime(G::ZZGenus)
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem})
is_elementary_with_prime(G::ZZGenus)
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem})

local Symbol

local_symbol(G::ZZGenus, p)

Representative(s)

quadratic_space(G::ZZGenus)
rational_representative(G::ZZGenus)
representative(G::ZZGenus)
representatives(G::ZZGenus)
mass(G::ZZGenus)
rescale(::ZZGenus, ::RationalUnion)

Embeddings and Representations

represents(G1::ZZGenus, G2::ZZGenus)

Local genus Symbols

ZZLocalGenus

Creation

genus(::ZZLat, ::IntegerUnion)
genus(::QQMatrix, ::IntegerUnion)

Attributes

prime(S::ZZLocalGenus)
iseven(S::ZZLocalGenus)
symbol(S::ZZLocalGenus, scale::Int)
hasse_invariant(S::ZZLocalGenus)
det(S::ZZLocalGenus)
dim(S::ZZLocalGenus)
rank(S::ZZLocalGenus)
excess(S::ZZLocalGenus)
signature(S::ZZLocalGenus)
oddity(S::ZZLocalGenus)
scale(S::ZZLocalGenus)
norm(S::ZZLocalGenus)
level(S::ZZLocalGenus)

Representative

representative(S::ZZLocalGenus)
gram_matrix(S::ZZLocalGenus)
rescale(S::ZZLocalGenus, a::RationalUnion)

Direct sums

direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus)

Embeddings/Representations

represents(G1::ZZLocalGenus, G2::ZZLocalGenus)