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 [5] 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.

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Creation of Genera

From an integral Lattice

# genus — Method.

genus(L::ZZLat) -> ZZGenus

Return the genus of the lattice L.

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From a gram matrix

# genus — Method.

genus(A::MatElem) -> ZZGenus

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

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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)

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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.

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Attributes of the genus

# dim — Method.

dim(G::ZZGenus) -> Int

Return the dimension of this genus.

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# rank — Method.

rank(G::ZZGenus) -> Int

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

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# 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.

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# det — Method.

det(G::ZZGenus) -> QQFieldElem

Return the determinant of this genus.

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# iseven — Method.

iseven(G::ZZGenus) -> Bool

Return if this genus is even.

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# is_definite — Method.

is_definite(G::ZZGenus) -> Bool

Return if this genus is definite.

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# level — Method.

level(G::ZZGenus) -> QQFieldElem

Return the level of this genus.

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

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# 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).

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# 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 $\{b(x,x)|x\in L\}$.

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# 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.

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# is_integral — Method.

is_integral(G::ZZGenus) -> Bool

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

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Discriminant group

discriminant_group(::ZZGenus)

Primary genera

# is_primary_with_prime — Method.

is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem

Given a genus of $\mathbb{Z}$-lattices G, return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that for unimodular genera, this function returns (true, 1). If the genus is not primary, the second return value is -1 by default.

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# is_primary — Method.

is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool

Given a genus of integral $\mathbb{Z}$-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.

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# is_elementary_with_prime — Method.

is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem

Given a genus of $\mathbb{Z}$-lattices G, return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see discriminant_group) is an elementary p-group for some prime number p. In case it is, p is also returned as second output.

Note that for unimodular genera, this function returns (true, 1). If the genus is not elementary, the second return value is -1 by default.

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# is_elementary — Method.

is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool

Given a genus of integral $\mathbb{Z}$-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.

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local Symbol

# local_symbol — Method.

local_symbol(G::ZZGenus, p) -> ZZLocalGenus

Return the local symbol at p.

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Representative(s)

# quadratic_space — Method.

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

Return the quadratic space defined by this genus.

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# rational_representative — Method.

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

Return the quadratic space defined by this genus.

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# representative — Method.

representative(G::ZZGenus) -> ZZLat

Compute a representative of this genus && cache it.

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# representatives — Method.

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

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

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# 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 $\sum _{i=1}^n\frac{1}{|O(L_i)|}$.

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# 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.

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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.

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Local genus Symbols

# ZZLocalGenus — Type.

ZZLocalGenus

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: [5] 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

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Creation

# genus — Method.

genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus

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

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# genus — Method.

genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus

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

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Attributes

# prime — Method.

prime(S::ZZLocalGenus) -> ZZRingElem

Return the prime p of this p-adic genus.

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# iseven — Method.

iseven(S::ZZLocalGenus) -> Bool

Return if the underlying p-adic lattice is even.

If p is odd, every lattice is even.

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# symbol — Method.

symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}

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

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# hasse_invariant — Method.

hasse_invariant(S::ZZLocalGenus) -> Int

Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is

$$\prod _{i<j}(a_i,a_j{)}_p$$

.

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# det — Method.

det(S::ZZLocalGenus) -> QQFieldElem

Return an rational representing the determinant of this genus.

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# dim — Method.

dim(S::ZZLocalGenus) -> Int

Return the dimension of this genus.

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# rank — Method.

rank(S::ZZLocalGenus) -> Int

Return the rank of (a representative of) S.

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# excess — Method.

excess(S::ZZLocalGenus) -> 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

[5] pp 370-371.

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# signature — Method.

signature(S::ZZLocalGenus) -> zzModRingElem

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

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# oddity — Method.

oddity(S::ZZLocalGenus) -> zzModRingElem

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

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# scale — Method.

scale(S::ZZLocalGenus) -> 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).

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# norm — Method.

norm(S::ZZLocalGenus) -> 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 $\{b(x,x)|x\in L\}$.

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# level — Method.

level(S::ZZLocalGenus) -> QQFieldElem

Return the maximal scale of a jordan component.

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Representative

# representative — Method.

representative(S::ZZLocalGenus) -> ZZLat

Return an integer lattice which represents this local genus.

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# gram_matrix — Method.

gram_matrix(S::ZZLocalGenus) -> MatElem

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

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# rescale — Method.

rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus

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

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Direct sums

# direct_sum — Method.

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

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

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Embeddings/Representations

# represents — Method.

represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> 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) $2^i(1+4\omega )\to {\mathfrak{L}}_{i+1}/{\mathfrak{l}}_{[i]}$.

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