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Genera for hermitian lattices

CurrentModule = Hecke
DocTestSetup = quote
    using Hecke
  end

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.

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det)
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
g2 = genus(L, p)

Attributes

length(::HermLocalGenus)
base_field(::HermLocalGenus)
prime(::HermLocalGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
length(g1)
base_field(g1)
prime(g1)

Invariants

scale(::HermLocalGenus, ::Int)
scale(::HermLocalGenus)
scales(::HermLocalGenus)
rank(::HermLocalGenus, ::Int)
rank(::HermLocalGenus)
ranks(::HermLocalGenus)
det(::HermLocalGenus, ::Int)
det(::HermLocalGenus)
dets(::HermLocalGenus)
discriminant(::HermLocalGenus, ::Int)
discriminant(::HermLocalGenus)
norm(::HermLocalGenus, ::Int)
norm(::HermLocalGenus)
norms(::HermLocalGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
g2 = genus(L, p);
scales(g2)
ranks(g2)
dets(g2)
norms(g2)
rank(g2), det(g2), discriminant(g2)

Predicates

is_ramified(::HermLocalGenus)
is_split(::HermLocalGenus)
is_inert(::HermLocalGenus)
is_dyadic(::HermLocalGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
is_ramified(g1), is_split(g1), is_inert(g1), is_dyadic(g1)

Local uniformizer

uniformizer(::HermLocalGenus)

Example

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
uniformizer(g1)

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(::HermLocalGenus, ::Int)
det_representative(::HermLocalGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
det_representative(g1)
det_representative(g1,2)

Gram matrices

gram_matrix(::HermLocalGenus, ::Int)
gram_matrix(::HermLocalGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
g2 = genus(L, p);
gram_matrix(g2)
gram_matrix(g2,1)

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 signatures signatures 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.

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
infp = infinite_places(E)
SEK = unique([r.base_field_place for r in infp if isreal(r.base_field_place) && !isreal(r)]);
length(SEK)
G1 = genus([g1], [(SEK[1], 1)])
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
G2 = genus(L)

Attributes

base_field(::HermGenus)
primes(::HermGenus)
signatures(::HermGenus)
rank(::HermGenus)
is_integral(::HermGenus)
local_symbols(::HermGenus)
scale(::HermGenus)
norm(::HermGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
G2 = genus(L);
base_field(G2)
primes(G2)
signatures(G2)
rank(G2)

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(::HermLat)

Example

using Hecke # hide
Qx, x = polynomial_ring(FlintQQ, "x");
f = x^2 - 2;
K, a = number_field(f, "a", cached = false);
Kt, t = polynomial_ring(K, "t");
g = t^2 + 1;
E, b = number_field(g, "b", cached = false);
D = matrix(E, 3, 3, [1, 0, 0, 0, 1, 0, 0, 0, 1]);
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])];
L = hermitian_lattice(E, gens, gram = D);
mass(L)

Representatives of a genus

representative(::HermLocalGenus)
Base.in(::HermLat, ::HermLocalGenus)
representative(::HermGenus)
Base.in(::HermLat, ::HermGenus)
representatives(::HermGenus)
genus_representatives(::HermLat)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
G1 = genus([g1], [(SEK[1], 1)]);
L1 = representative(g1)
L1 in g1
L2 = representative(G1)
L2 in G1, L2 in g1
length(genus_representatives(L1))
length(representatives(G1))

Sum of genera

direct_sum(::HermLocalGenus, ::HermLocalGenus)
direct_sum(::HermGenus, ::HermGenus)

Examples

using Hecke # hide
Qx, x = QQ["x"];
K, a = number_field(x^2 - 2, "a");
Kt, t  = K["t"];
E, b = number_field(t^2 - a, "b");
OK = maximal_order(K);
p = prime_decomposition(OK, 2)[1][1];
g1 = genus(HermLat, E, p, [(0, 1, 1, 0), (2, 2, -1, 1)], type = :det);
SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
G1 = genus([g1], [(SEK[1], 1)]);
D = matrix(E, 3, 3, [5//2*a - 4, 0, 0, 0, a, a, 0, a, -4*a + 8]);
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])];
L = hermitian_lattice(E, gens, gram = D);
g2 = genus(L, p);
G2 = genus(L);
direct_sum(g1, g2)
direct_sum(G1, G2)

Enumeration of genera

hermitian_local_genera(E, p, ::Int, ::Int, ::Int, ::Int)
hermitian_genera(::Hecke.RelSimpleNumField, ::Int, ::Dict{InfPlc, Int}, ::Union{Hecke.RelNumFieldOrderIdeal, Hecke.RelNumFieldOrderFractionalIdeal})

Examples

using Hecke # hide
K, a = cyclotomic_real_subfield(8, "a");
Kt, t = K["t"];
E, b = number_field(t^2 - a * t + 1);
p = prime_decomposition(maximal_order(K), 2)[1][1];
hermitian_local_genera(E, p, 4, 2, 0, 4)
SEK = unique([restrict(r, K) for r in infinite_places(E) if isreal(restrict(r, K)) && !isreal(r)]);
hermitian_genera(E, 3, Dict(SEK[1] => 1, SEK[2] => 1), 30 * maximal_order(E))

Rescaling

rescale(g::HermLocalGenus, a::Union{FieldElem, RationalUnion})
rescale(G::HermGenus, a::Union{FieldElem, RationalUnion})