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 maximal order
    of relative number field with defining polynomial t^2 - a
      over number field of degree 2 over QQ
  with pseudo-basis
    (1, <1>//1)
    (b, <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 maximal order
    of relative number field with defining polynomial t^2 - a
      over number field of degree 2 over QQ
  with pseudo-basis
    (1, <1>//1)
    (b, <1>//1)
Prime ideal: <2, a>
Jordan blocks (scale, rank, det, norm):
  (-2, 1, +, -1)
  (2, 2, +, 1)

Attributes

length — Method

length(g::HermLocalGenus) -> Int

Given a local genus symbol g for hermitian lattices, return the number of Jordan blocks of g.