Genera for hermitian lattices
Local genus symbols
Definition 8.3.1 ([Kir16]) Let $L$ be a hermitian lattice over $E/K$ and let $\mathfrak p$ be a prime ideal of $\mathcal O_K$. Let $\mathfrak P$ be the largest ideal of $\mathcal O_E$ over $\mathfrak p$ being invariant under the involution of $E$. We suppose that we are given a Jordan decomposition
\[ L_{\mathfrak p} = \perp_{i=1}^tL_i\]
where the Jordan block $L_i$ is $\mathfrak P^{s_i}$-modular for $1 \leq i \leq t$, for a strictly increasing sequence of integers $s_1 < \ldots < s_t$. In particular, $\mathfrak s(L_i) = \mathfrak P^{s_i}$. Then, the local genus symbol $g(L, \mathfrak p)$ of $L_{\mathfrak p}$ is defined to be:
- if $\mathfrak p$ is good, i.e. non ramified and non dyadic,
\[ g(L, \mathfrak p) := [(s_1, r_1, d_1), \ldots, (s_t, r_t, d_t)]\]
where $d_i = 1$ if the determinant (resp. discriminant) of $L_i$ is a norm in $K_{\mathfrak p}^{\times}$, and $d_i = -1$ otherwise, and $r_i := \text{rank}(L_i)$ for all i;
- if $\mathfrak p$ is bad,
\[ g(L, \mathfrak p) := [(s_1, r_1, d_1, n_1), \ldots, (s_t, r_t, d_t, n_t)]\]
where for all i, $n_i := \text{ord}_{\mathfrak p}(\mathfrak n(L_i))$
Note that we define the scale and the norm of the lattice $L_i$ ($1 \leq i \leq n$) defined over the extension of local fields $E_{\mathfrak P}/K_{\mathfrak p}$ similarly to the ones of $L$, by extending by continuity the sesquilinear form of the ambient space of $L$ to the completion. Regarding the determinant (resp. discriminant), it is defined as the determinant of the Gram matrix associated to a basis of $L_i$ relatively to the extension of the sesquilinear form (resp. $(-1)^{(m(m-1)/2}$ times the determinant, where $m$ is the rank of $L_i$).
We call any tuple in $g := g(L, \mathfrak p) = [g_1, \ldots, g_t]$ a Jordan block of $g$ since it corresponds to invariants of a Jordan block of the completion of the lattice $L$ at $\mathfrak p$. For any such block $g_i$, we call respectively $s_i, r_i, d_i, n_i$ the scale, the rank, the determinant class (resp. discriminant class) and the norm of $g_i$. Note that the norm is necessary only when the prime ideal is bad.
We say that two hermitian lattices $L$ and $L'$ over $E/K$ are in the same local genus at $\mathfrak p$ if $g(L, \mathfrak p) = g(L', \mathfrak p)$.
Creation of local genus symbols
There are two ways of creating a local genus symbol for hermitian lattices:
- either abstractly, by choosing the extension $E/K$, the prime ideal $\mathfrak p$ of $\mathcal O_K$, the Jordan blocks
data
and the type of the $d_i$'s (either determinant class:det
or discriminant class:disc
);
genus(HermLat, E::NumField, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, data::Vector; type::Symbol = :det,
check::Bool = false)
-> HermLocalGenus
- or by constructing the local genus symbol of the completion of a hermitian lattice $L$ over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$.
genus(L::HermLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> HermLocalGenus
Examples
We will construct two examples for the rest of this section. Note that the prime chosen here is bad.
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det)
Local genus symbol for hermitian lattices over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>) Prime ideal: <2, a> Jordan blocks (scale, rank, det, norm): (0, 1, +, 0) (2, 2, -, 1)
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> g2 = genus(L, p)
Local genus symbol for hermitian lattices over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>) Prime ideal: <2, a> Jordan blocks (scale, rank, det, norm): (-2, 1, +, -1) (2, 2, +, 1)
Attributes
length
— Methodlength(g::HermLocalGenus) -> Int
Given a local genus symbol g
for hermitian lattices, return the number of Jordan blocks of g
.
base_field
— Methodbase_field(g::HermLocalGenus) -> NumField
Given a local genus symbol g
for hermitian lattices over $E/K$, return E
.
prime
— Methodprime(g::HermLocalGenus) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return $\mathfrak p$.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> length(g1)
2
julia> base_field(g1)
Relative number field with defining polynomial t^2 - a over number field with defining polynomial x^2 - 2 over rational field
julia> prime(g1)
<2, a> Norm: 2 Minimum: 2 basis_matrix [2 0; 0 1] two normal wrt: 2
Invariants
scale
— Methodscale(g::HermLocalGenus, i::Int) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime $\mathfrak p$ of $\mathcal O_K$, return the $\mathfrak P$-valuation of the scale of the i
th Jordan block of g
, where $\mathfrak P$ is a prime ideal of $\mathcal O_E$ lying over $\mathfrak p$.
scale
— Methodscale(g::HermLocalGenus) -> AbsSimpleNumFieldOrderFractionalIdeal
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime $\mathfrak p$ of $\mathcal O_K$, return the scale of the Jordan block of minimum $\mathfrak P$-valuation, where $\mathfrak{P}$ is a prime ideal of $\mathcal O_E$ lying over $\mathfrak p$.
scales
— Methodscales(g::HermLocalGenus) -> Vector{Int}
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime $\mathfrak p$ of $\mathcal O_K$, return the $\mathfrak P$-valuation of the scales of the Jordan blocks of g
, where $\mathfrak P$ is a prime ideal of $\mathcal O_E$ lying over $\mathfrak p$.
rank
— Methodrank(g::HermLocalGenus, i::Int) -> Int
Given a local genus symbol g
for hermitian lattices, return the rank of the i
th Jordan block of g
.
rank
— Methodrank(g::HermLocalGenus) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return the rank of any hermitian lattice whose $\mathfrak p$-adic completion has local genus symbol g
.
ranks
— Methodranks(g::HermLocalGenus) -> Vector{Int}
Given a local genus symbol g
for hermitian lattices, return the ranks of the Jordan blocks of g
.
det
— Methoddet(g::HermLocalGenus, i::Int) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$, return the determinant of the i
th Jordan block of g
.
The returned value is $1$ or $-1$ depending on whether the determinant is a local norm in K
.
det
— Methoddet(g::HermLocalGenus) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return the determinant of a hermitian lattice whose $\mathfrak p$-adic completion has local genus symbol g
.
The returned value is $1$ or $-1$ depending on whether the determinant is a local norm in K
.
dets
— Methoddets(g::HermLocalGenus) -> Vector{Int}
Given a local genus symbol g
for hermitian lattices over $E/K$, return the determinants of the Jordan blocks of g
.
The returned values are $1$ or $-1$ depending on whether the respective determinants are are local norms in K
.
discriminant
— Methoddiscriminant(g::HermLocalGenus, i::Int) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$, return the discriminant of the i
th Jordan block of g
.
The returned value is $1$ or $-1$ depending on whether the discriminant is a local norm in K
.
discriminant
— Methoddiscriminant(g::HermLocalGenus) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return the discriminant of a hermitian lattice whose $\mathfrak p$-adic completion has local genus symbol g
.
The returned value is $1$ or $-1$ depending on whether the discriminant is a local norm in K
.
norm
— Methodnorm(g::HermLocalGenus, i::Int) -> Int
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return the $\mathfrak p$-valuation of the norm of the i
th Jordan block of g
.
norm
— Methodnorm(g::HermLocalGenus) -> AbsSimpleNumFieldOrderFractionalIdeal
Return the norm of g
, i.e. the norm of any of its representatives.
Given a local genus symbol g
of hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, it norm is computed as the norm of the Jordan block of minimum $\mathfrak p$-valuation.
norms
— Methodnorms(g::HermLocalGenus) -> Vector{Int}
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return the $\mathfrak p$-valuations of the norms of the Jordan blocks of g
.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> g2 = genus(L, p);
julia> scales(g2)
2-element Vector{Int64}: -2 2
julia> ranks(g2)
2-element Vector{Int64}: 1 2
julia> dets(g2)
2-element Vector{Int64}: 1 1
julia> norms(g2)
2-element Vector{Int64}: -1 1
julia> rank(g2), det(g2), discriminant(g2)
(3, 1, -1)
Predicates
is_ramified
— Methodis_ramified(g::HermLocalGenus) -> Bool
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return whether $\mathfrak p$ is ramified in $\mathcal O_E$.
is_split
— Methodis_split(g::HermLocalGenus) -> Bool
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return whether $\mathfrak p$ is split in $\mathcal O_E$.
is_inert
— Methodis_inert(g::HermLocalGenus) -> Bool
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return whether $\mathfrak p$ is inert in $\mathcal O_E$.
is_dyadic
— Methodis_dyadic(g::HermLocalGenus) -> Bool
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return whether $\mathfrak p$ is dyadic.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> is_ramified(g1), is_split(g1), is_inert(g1), is_dyadic(g1)
(true, false, false, true)
Local uniformizer
uniformizer
— Methoduniformizer(g::HermLocalGenus) -> NumFieldElem
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return a generator for the largest ideal of $\mathcal O_E$ containing $\mathfrak p$ and invariant under the action of the non-trivial involution of E
.
Example
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> uniformizer(g1)
-a
Determinant representatives
Let $g$ be a local genus symbol for hermitian lattices. Its determinant class, or the determinant class of its Jordan blocks, are given by $\pm 1$, depending on whether the determinants are local norms or not. It is possible to get a representative of this determinant class in terms of powers of the uniformizer of $g$.
det_representative
— Methoddet_representative(g::HermLocalGenus, i::Int) -> NumFieldElem
Given a local genus symbol g
for hermitian lattices over $E/K$, return a representative of the norm class of the determinant of the i
th Jordan block of g
in $K^{\times}$.
det_representative
— Methoddet_representative(g::HermLocalGenus) -> NumFieldElem
Given a local genus symbol g
for hermitian lattices over $E/K$, return a representative of the norm class of the determinant of g
in $K^{\times}$.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> det_representative(g1)
-8*a + 10
julia> det_representative(g1,2)
-8*a + 10
Gram matrices
gram_matrix
— Methodgram_matrix(g::HermLocalGenus, i::Int) -> MatElem
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return a Gram matrix M
of the i
th Jordan block of g
, with coefficients in E
. M
is such that any hermitian lattice over $E/K$ with Gram matrix M
satisfies that the local genus symbol of its completion at $\mathfrak p$ is equal to the i
th Jordan block of g
.
gram_matrix
— Methodgram_matrix(g::HermLocalGenus) -> MatElem
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return a Gram matrix M
of g
, with coefficients in E
.M
is such that any hermitian lattice over $E/K$ with Gram matrix M
satisfies that the local genus symbol of its completion at $\mathfrak p$ is g
.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> g2 = genus(L, p);
julia> gram_matrix(g2)
[-3//2*a - 4 0 0] [ 0 a a] [ 0 a 4*a + 8]
julia> gram_matrix(g2,1)
[-3//2*a - 4]
Global genus symbols
Let $L$ be a hermitian lattice over $E/K$. Let $P(L)$ be the set of all prime ideals of $\mathcal O_K$ which are bad (ramified or dyadic), which are dividing the scale of $L$ or which are dividing the volume of $L$. Let $S(E/K)$ be the set of real infinite places of $K$ which split into complex places in $E$. We define the global genus symbol $G(L)$ of $L$ to be the datum consisting of the local genus symbols of $L$ at each prime of $P(L)$ and the signatures (i.e. the negative index of inertia) of the Gram matrix of the rational span of $L$ at each place in $S(E/K)$.
Note that prime ideals in $P(L)$ which don't ramify correspond to those for which the corresponding completions of $L$ are not unimodular.
We say that two lattice $L$ and $L'$ over $E/K$ are in the same genus, if $G(L) = G(L')$.
Creation of global genus symbols
Similarly, there are two ways of constructing a global genus symbol for hermitian lattices:
- either abstractly, by choosing the extension $E/K$, the set of local genus symbols
S
and the signaturessignatures
at the places in $S(E/K)$. Note that this requires the given invariants to satisfy the product formula for Hilbert symbols.
genus(S::Vector{HermLocalGenus}, signatures) -> HermGenus
Here signatures
can be a dictionary with keys the infinite places and values the corresponding signatures, or a collection of tuples of the type (::InfPlc, ::Int)
;
- or by constructing the global genus symbol of a given hermitian lattice $L$.
genus(L::HermLat) -> HermGenus
Examples
As before, we will construct two different global genus symbols for hermitian lattices, which we will use for the rest of this section.
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> infp = infinite_places(E)
3-element Vector{InfPlc{Hecke.RelSimpleNumField{AbsSimpleNumFieldElem}, RelSimpleNumFieldEmbedding{AbsSimpleNumFieldEmbedding, Hecke.RelSimpleNumField{AbsSimpleNumFieldElem}}}}: Infinite place corresponding to (Complex embedding corresponding to root -1.19 of relative number field) Infinite place corresponding to (Complex embedding corresponding to root 1.19 of relative number field) Infinite place corresponding to (Complex embedding corresponding to root 0.00 + 1.19 * i of relative number field)
julia> SEK = unique([r.base_field_place for r in infp if isreal(r.base_field_place) && !isreal(r)]);
ERROR: type InfPlc has no field base_field_place
julia> length(SEK)
ERROR: UndefVarError: `SEK` not defined
julia> G1 = genus([g1], [(SEK[1], 1)])
ERROR: UndefVarError: `SEK` not defined
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> G2 = genus(L)
Genus symbol for hermitian lattices over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>) Signature: infinite place corresponding to (Complex embedding of number field) => 2 Local symbols: <2, a> => (-2, 1, +, -1)(2, 2, +, 1) <7, a + 4> => (0, 1, +)(1, 2, +)
Attributes
base_field
— Methodbase_field(G::HermGenus) -> NumField
Given a global genus symbol G
for hermitian lattices over $E/K$, return E
.
primes
— Methodprimes(G::HermGenus) -> Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}
Given a global genus symbol G
for hermitian lattices over $E/K$, return the list of prime ideals of $\mathcal O_K$ at which G
has a local genus symbol.
signatures
— Methodsignatures(G::HermGenus) -> Dict{InfPlc, Int}
Given a global genus symbol G
for hermitian lattices over $E/K$, return the signatures at the infinite places of K
. For each real place, it is given by the negative index of inertia of the Gram matrix of the rational span of a hermitian lattice whose global genus symbol is G
.
The output is given as a dictionary with keys the infinite places of K
and value the corresponding signatures.
rank
— Methodrank(G::HermGenus) -> Int
Return the rank of any hermitian lattice with global genus symbol G
.
is_integral
— Methodis_integral(G::HermGenus) -> Bool
Return whether G
defines a genus of integral hermitian lattices.
local_symbols
— Methodlocal_symbols(G::HermGenus) -> Vector{HermLocalGenus}
Given a global genus symbol of hermitian lattices, return its associated local genus symbols.
scale
— Methodscale(G::HermGenus) -> AbsSimpleNumFieldOrderFractionalIdeal
Return the scale ideal of any hermitian lattice with global genus symbol G
.
norm
— Methodnorm(G::HermGenus) -> AbsSimpleNumFieldOrderFractionalIdeal
Return the norm ideal of any hermitian lattice with global genus symbol G
.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> G2 = genus(L);
julia> base_field(G2)
Relative number field with defining polynomial t^2 - a over number field with defining polynomial x^2 - 2 over rational field
julia> primes(G2)
2-element Vector{AbsSimpleNumFieldOrderIdeal}: <2, a> Norm: 2 Minimum: 2 basis_matrix [2 0; 0 1] two normal wrt: 2 <7, a + 4> Norm: 7 Minimum: 7 basis_matrix [7 0; 4 1] two normal wrt: 7
julia> signatures(G2)
Dict{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}, Int64} with 1 entry: Infinite place corresponding to (Complex embedding corresponding to -1.4… => 2
julia> rank(G2)
3
Mass
Definition 4.2.1 [Kir16] Let $L$ be a hermitian lattice over $E/K$, and suppose that $L$ is definite. In particular, the automorphism group of $L$ is finite. Let $L_1, \ldots, L_n$ be a set of representatives of isometry classes in the genus of $L$. This means that if $L'$ is a lattice over $E/K$ in the genus of $L$ (i.e. they are in the same genus), then $L'$ is isometric to one of the $L_i$'s, and these representatives are pairwise non-isometric. Then we define the mass of the genus $G(L)$ of $L$ to be
\[ \text{mass}(G(L)) := \sum_{i=1}^n\frac{1}{\#\text{Aut}(L_i)}.\]
Note that since $L$ is definite, any lattice in the genus of $L$ is also definite, and the definition makes sense.
mass
— Methodmass(L::HermLat) -> QQFieldElem
Given a definite hermitian lattice L
, return the mass of its genus.
Example
julia> Qx, x = polynomial_ring(QQ, "x");
julia> f = x^2 - 2;
julia> K, a = number_field(f, "a", cached = false);
julia> Kt, t = polynomial_ring(K, "t");
julia> g = t^2 + 1;
julia> E, b = number_field(g, "b", cached = false);
julia> D = matrix(E, 3, 3, [1, 0, 0, 0, 1, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [(-3*a + 7)*b + 3*a, (5//2*a - 1)*b - 3//2*a + 4, 0]), map(E, [(3004*a - 4197)*b - 3088*a + 4348, (-1047//2*a + 765)*b + 5313//2*a - 3780, (-a - 1)*b + 3*a - 1]), map(E, [(728381*a - 998259)*b + 3345554*a - 4653462, (-1507194*a + 2168244)*b - 1507194*a + 2168244, (-5917//2*a - 915)*b - 4331//2*a - 488])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> mass(L)
1//1024
Representatives of a genus
representative
— Methodrepresentative(g::HermLocalGenus) -> HermLat
Given a local genus symbol g
for hermitian lattices over $E/K$ at a prime ideal $\mathfrak p$ of $\mathcal O_K$, return a hermitian lattice over $E/K$ whose completion at $\mathfrak p$ admits g
as local genus symbol.
in
— Methodin(L::HermLat, g::HermLocalGenus) -> Bool
Return whether g
and the local genus symbol of the completion of the hermitian lattice L
at prime(g)
agree. Note that L
being in g
requires both L
and g
to be defined over the same extension $E/K$.
representative
— Methodrepresentative(G::HermGenus) -> HermLat
Given a global genus symbol G
for hermitian lattices over $E/K$, return a hermitian lattice over $E/K$ which admits G
as global genus symbol.
in
— Methodin(L::HermLat, G::HermGenus) -> Bool
Return whether G
and the global genus symbol of the hermitian lattice L
agree.
representatives
— Methodrepresentatives(G::HermGenus) -> Vector{HermLat}
Given a global genus symbol G
for hermitian lattices, return representatives for the isometry classes of hermitian lattices in G
.
genus_representatives
— Methodgenus_representatives(L::HermLat; max = inf, use_auto = true,
use_mass = false)
-> Vector{HermLat}
Return representatives for the isometry classes in the genus of the hermitian lattice L
. At most max
representatives are returned.
If L
is definite, the use of the automorphism group of L
is enabled by default. It can be disabled by use_auto = false
. In the case where L
is indefinite, the entry use_auto
has no effect. The computation of the mass can be enabled by use_mass = true
.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
julia> G1 = genus([g1], [(SEK[1], 1)]);
julia> L1 = representative(g1)
Hermitian lattice of rank 3 and degree 3 over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>)
julia> L1 in g1
true
julia> L2 = representative(G1)
Hermitian lattice of rank 3 and degree 3 over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>)
julia> L2 in G1, L2 in g1
(true, true)
julia> length(genus_representatives(L1))
1
julia> length(representatives(G1))
1
Sum of genera
direct_sum
— Methoddirect_sum(g1::HermLocalGenus, g2::HermLocalGenus) -> HermLocalGenus
Given two local genus symbols g1
and g2
for hermitian lattices over $E/K$ at the same prime ideal $\mathfrak p$ of $\mathcal O_K$, return their direct sum. It corresponds to the local genus symbol of the $\mathfrak p$-adic completion of the direct sum of respective representatives of g1
and g2
.
direct_sum
— Methoddirect_sum(G1::HermGenus, G2::HermGenus) -> HermGenus
Given two global genus symbols G1
and G2
for hermitian lattices over $E/K$, return their direct sum. It corresponds to the global genus symbol of the direct sum of respective representatives of G1
and G2
.
Examples
julia> Qx, x = QQ["x"];
julia> K, a = number_field(x^2 - 2, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a, "b");
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 2)[1][1];
julia> g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
julia> SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
julia> G1 = genus([g1], [(SEK[1], 1)]);
julia> D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [1, 0, 0]), map(E, [a, 0, 0]), map(E, [b, 0, 0]), map(E, [a*b, 0, 0]), map(E, [0, 1, 0]), map(E, [0, a, 0]), map(E, [0, b, 0]), map(E, [0, a*b, 0]), map(E, [0, 0, 1]), map(E, [0, 0, a]), map(E, [0, 0, b]), map(E, [0, 0, a*b])];
julia> L = hermitian_lattice(E, gens, gram = D);
julia> g2 = genus(L, p);
julia> G2 = genus(L);
julia> direct_sum(g1, g2)
Local genus symbol for hermitian lattices over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>) Prime ideal: <2, a> Jordan blocks (scale, rank, det, norm): (-2, 1, +, -1) (0, 1, +, 0) (2, 4, -, 1)
julia> direct_sum(G1, G2)
Genus symbol for hermitian lattices over relative maximal order of Relative number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (b, 1//1 * <1, 1>) Signature: infinite place corresponding to (Complex embedding of number field) => 3 Local symbols: <2, a> => (-2, 1, +, -1)(0, 1, +, 0)(2, 4, -, 1) <7, a + 4> => (0, 4, +)(1, 2, +)
Enumeration of genera
hermitian_local_genera
— Methodhermitian_local_genera(E::NumField, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, rank::Int,
det_val::Int, min_scale::Int, max_scale::Int)
-> Vector{HermLocalGenus}
Return all local genus symbols for hermitian lattices over the algebra E
, with base field $K$, at the prime idealp
of $\mathcal O_K$. Each of them has rank equal to rank
, scale $\mathfrak P$-valuations bounded between min_scale
and max_scale
and determinant p
-valuations equal to det_val
, where $\mathfrak P$ is a prime ideal of $\mathcal O_E$ lying above p
.
hermitian_genera
— Methodhermitian_genera(E::NumField, rank::Int,
signatures::Dict{InfPlc, Int},
determinant::Union{Hecke.RelNumFieldOrderIdeal, Hecke.RelNumFieldOrderFractionalIdeal};
min_scale::Union{Hecke.RelNumFieldOrderIdeal, Hecke.RelNumFieldOrderFractionalIdeal} = is_integral(determinant) ? inv(1*order(determinant)) : determinant,
max_scale::Union{Hecke.RelNumFieldOrderIdeal, Hecke.RelNumFieldOrderFractionalIdeal} = is_integral(determinant) ? determinant : inv(1*order(determinant)))
-> Vector{HermGenus}
Return all global genus symbols for hermitian lattices over the algebraE
with rank rank
, signatures given by signatures
, scale bounded by max_scale
and determinant class equal to determinant
.
If max_scale == nothing
, it is set to be equal to determinant
.
Examples
julia> K, a = cyclotomic_real_subfield(8, "a");
julia> Kt, t = K["t"];
julia> E, b = number_field(t^2 - a * t + 1);
julia> p = prime_decomposition(maximal_order(K), 2)[1][1];
julia> hermitian_local_genera(E, p, 4, 2, 0, 4)
15-element Vector{HermLocalGenus{Hecke.RelSimpleNumField{AbsSimpleNumFieldElem}, AbsSimpleNumFieldOrderIdeal}}: Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers Local genus symbol for hermitian lattices over the 2-adic integers
julia> SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
julia> hermitian_genera(E, 3, Dict(SEK[1] => 1, SEK[2] => 1), 30 * maximal_order(E))
6-element Vector{HermGenus{Hecke.RelSimpleNumField{AbsSimpleNumFieldElem}, AbsSimpleNumFieldOrderIdeal, HermLocalGenus{Hecke.RelSimpleNumField{AbsSimpleNumFieldElem}, AbsSimpleNumFieldOrderIdeal}, Dict{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}, Int64}}}: Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>) Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>) Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>) Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>) Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>) Genus symbol for hermitian lattices of rank 3 over relative maximal order of Relative number field with pseudo-basis (1, 1//1 * <1, 1>) (_$, 1//1 * <1, 1>)
Rescaling
rescale
— Methodrescale(g::HermLocalGenus, a::Union{FieldElem, RationalUnion})
-> HermLocalGenus
Given a local genus symbol G
of hermitian lattices and an element a
lying in the base field E
of g
, return the local genus symbol at the prime ideal p
associated to g
of any representative of g
rescaled by a
.
rescale
— Methodrescale(G::HermGenus, a::Union{FieldElem, RationalUnion}) -> HermGenus
Given a global genus symbol G
of hermitian lattices and an element a
lying in the base field E
of G
, return the global genus symbol of any representative of G
rescaled by a
.