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Genera of Integer Lattices

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 J. H. Conway, N. J. A. Sloane (1999) Chapter 15 for the definitions. Note that genera for non-integral lattices are supported.

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

ZZGenus — Type
ZZGenus

A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.

Creation of Genera

From an integral Lattice

genus — Method
genus(L::ZZLat) -> ZZGenus

Return the genus of the lattice L.

From a gram matrix

genus — Method
genus(A::MatElem) -> ZZGenus

Return the genus of a $\mathbb Z$-lattice with gram matrix A.

Enumeration of genus symbols

integer_genera — Method
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
       min_scale::RationalUnion = min(one(QQ), QQ(abs(determinant))),
       max_scale::RationalUnion = max(one(QQ), QQ(abs(determinant))),
       even=false)                                         -> Vector{ZZGenus}

Return a list of all genera with the given conditions. Genera of non-integral $\mathbb Z$-lattices are also supported.

Arguments

  • sig_pair: a pair of non-negative integers giving the signature
  • determinant: a rational number; the sign is ignored
  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))
  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))
  • even: boolean; if set to true, return only the even genera (default: false)

From other genus symbols

direct_sum — Method
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus

Return the genus of the direct sum of G1 and G2.

The direct sum is defined via representatives.

Attributes of the genus

dim — Method
dim(G::ZZGenus) -> Int

Return the dimension of this genus.

rank — Method
rank(G::ZZGenus) -> Int

Return the rank of a (representative of) the genus G.

signature — Method
signature(G::ZZGenus) -> Int

Return the signature of this genus.

The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.

det — Method
det(G::ZZGenus) -> QQFieldElem

Return the determinant of this genus.

iseven — Method
iseven(G::ZZGenus) -> Bool

Return if this genus is even.

is_definite — Method
is_definite(G::ZZGenus) -> Bool

Return if this genus is definite.

level — Method
level(G::ZZGenus) -> QQFieldElem

Return the level of this genus.

This is the denominator of the inverse gram matrix of a representative.

scale — Method
scale(G::ZZGenus) -> QQFieldElem

Return the scale of this genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

norm — Method
norm(G::ZZGenus) -> QQFieldElem

Return the norm of this genus.

Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by $\{b(x,x) | x \in L\}$.

primes — Method
primes(G::ZZGenus) -> Vector{ZZRingElem}

Return the list of primes of the local symbols of G.

Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.

is_integral — Method
is_integral(G::ZZGenus) -> Bool

Return whether G is a genus of integral $\mathbb Z$-lattices.

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 — Method
local_symbol(G::ZZGenus, p) -> LocalZZGenus

Return the local symbol at p.

Representative(s)

quadratic_space — Method
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

rational_representative — Method
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

representative — Method
representative(G::ZZGenus) -> ZZLat

Compute a representative of this genus && cache it.

representatives — Method
representatives(G::ZZGenus) -> Vector{ZZLat}

Return a list of representatives of the isometry classes in this genus.

mass — Method
mass(G::ZZGenus) -> QQFieldElem

Return the mass of this genus.

The genus must be definite. Let L_1, ... L_n be a complete list of representatives of the isometry classes in this genus. Its mass is defined as $\sum_{i=1}^n \frac{1}{|O(L_i)|}$.

rescale — Method
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus

Given a genus symbol G of $\mathbb Z$-lattices, return the genus symbol of any representative of G rescaled by a.

Embeddings and Representations

represents — Method
represents(G1::ZZGenus, G2::ZZGenus) -> Bool

Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.

Local genus Symbols

LocalZZGenus — Type
LocalZZGenus

Local genus symbol over a p-adic ring.

The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where

  • m = valuation of the component
  • n = rank of A
  • d = det(A) \in \{1,u\} for a normalized quadratic non-residue u.

The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where

  • m = valuation of the component
  • n = rank of A
  • d = det(A) in {1,3,5,7}
  • s = 0 (or 1) if even (or odd)
  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A

The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.

Reference: J. H. Conway, N. J. A. Sloane (1999) Chapter 15, Section 7.

Arguments

  • prime: a prime number
  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers

Creation

genus — Method
genus(L::ZZLat, p) -> LocalZZGenus

Return the local genus symbol of L at the prime p.

genus — Method
genus(A::MatElem, p) -> LocalZZGenus

Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.

Attributes

prime — Method
prime(S::LocalZZGenus) -> ZZRingElem

Return the prime p of this p-adic genus.

iseven — Method
iseven(S::LocalZZGenus) -> Bool

Return if the underlying p-adic lattice is even.

If p is odd, every lattice is even.

symbol — Method
symbol(S::LocalZZGenus, scale::Int) -> Vector{Int}

Return a copy of the underlying lists of integers for the Jordan block of the given scale

hasse_invariant — Method
hasse_invariant(S::LocalZZGenus) -> Int

Return the Hasse invariant of a representative. If the representative is diagonal (a1, ... , an) Then the Hasse invariant is

\[\prod_{i < j}(a_i, a_j)_p\]

.

det — Method
det(S::LocalZZGenus) -> QQFieldElem

Return an rational representing the determinant of this genus.

dim — Method
dim(S::LocalZZGenus) -> Int

Return the dimension of this genus.

rank — Method
rank(S::LocalZZGenus) -> Int

Return the rank of (a representative of) S.

excess — Method
excess(S::LocalZZGenus) -> zzModRingElem

Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.

When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.

Reference

J. H. Conway, N. J. A. Sloane (1999) pp 370-371.

signature — Method
signature(S::LocalZZGenus) -> zzModRingElem

Return the $p$-signature of this $p$-adic form.

oddity — Method
oddity(S::LocalZZGenus) -> zzModRingElem

Return the oddity of this even form. The oddity is also called the $2$-signature

scale — Method
scale(S::LocalZZGenus) -> QQFieldElem

Return the scale of this local genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

norm — Method
norm(S::LocalZZGenus) -> QQFieldElem

Return the norm of this local genus.

Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by $\{b(x,x) | x \in L\}$.

level — Method
level(S::LocalZZGenus) -> QQFieldElem

Return the maximal scale of a jordan component.

Representative

representative — Method
representative(S::LocalZZGenus) -> ZZLat

Return an integer lattice which represents this local genus.

gram_matrix — Method
gram_matrix(S::LocalZZGenus) -> MatElem

Return a gram matrix of some representative of this local genus.

rescale — Method
rescale(G::LocalZZGenus, a::RationalUnion) -> LocalZZGenus

Given a local genus symbol G of $\mathbb Z$-lattices, return the local genus symbol of any representative of G rescaled by a.

Direct sums

direct_sum — Method
direct_sum(S1::LocalZZGenus, S2::LocalZZGenus) -> LocalZZGenus

Return the local genus of the direct sum of two representatives.

Embeddings/Representations

represents — Method
represents(g1::LocalZZGenus, g2::LocalZZGenus) -> Bool

Return whether g1 represents g2.

Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for p == 2 there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) $2^i(1+4\omega) \to \mathfrak{L}_{i+1}/\mathfrak{l}_{[i]}$.