Orders

Orders, that is, unitary subrings that are free $\mathbf{Z}$-modules of rank equal to the degree of the number field, are at the core of the arithmetic of number fields. In Hecke, orders are always represented using the module structure, be it the $\mathbf{Z}$-module structure for orders of absolute numbers fields, or the structure as a module over the maximal order of the base field in the case of relative number fields. In this chapter we mainly deal with orders of absolute fields. However, many functions apply in same way to relative extensions. There are more general definitions of orders in number fields available, but those are (currently) not implemented in Hecke.

Among all orders in a fixed field, there is a unique maximal order, called the maximal order, or ring of integers of the number field. It is well known that this is the only order that is a Dedekind domain, hence has a rich ideal structure as well. The maximal order is also the integral closure of $\mathbf{Z}$ in the number field and can also be interpreted as a normalization of any other order.

Creation and basic properties

Order — Method

Order(B::Vector{nf_elem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> NfOrd
Order(K::AnticNumberField, B::Vector{nf_elem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> NfOrd

Returns the order generated by $B$. If check is set, it is checked whether $B$ defines an order, in particular the integrality of the elements is checked by computing minimal polynomials. If isbasis is set, then elements are assumed to form a $\mathbf{Z}$-basis.

Order — Method

Order(K::AnticNumberField, A::FakeFmpqMat; check::Bool = true) -> NfOrd

Returns the order which has basis matrix $A$ with respect to the power basis of $K$. If check is set, it is checked whether $A$ defines an order.

Order — Method

Order(K::AnticNumberField, A::ZZMatrix, check::Bool = true) -> NfOrd

Returns the order which has basis matrix $A$ with respect to the power basis of $K$. If check is set, it is checked whether $A$ defines an order.

Order(A::AbsAlgAss{<: NumFieldElem}, M::PMat{<: NumFieldElem, T})
  -> AlgAssRelOrd

Returns the order of $A$ with basis pseudo-matrix $M$.

EquationOrder — Method

EquationOrder(K::number_field) -> NumFieldOrd
equation_order(K::number_field) -> NumFieldOrd

Returns the equation order of the number field $K$.

MaximalOrder — Method

MaximalOrder(K::NumField{QQFieldElem}; discriminant::ZZRingElem, ramified_primes::Vector{ZZRingElem}) -> NfAbsOrd

Returns the maximal order of $K$. Additional information can be supplied if they are already known, as the ramified primes or the discriminant of the maximal order.

Example

julia> Qx, x = FlintQQ["x"];
julia> K, a = number_field(x^3 + 2, "a");
julia> O = MaximalOrder(K);

MaximalOrder — Method

MaximalOrder(O::NfAbsOrd; index_divisors::Vector{ZZRingElem}, discriminant::ZZRingElem, ramified_primes::Vector{ZZRingElem}) -> NfAbsOrd

Returns the maximal order of the number field that contains $O$. Additional information can be supplied if they are already known, as the ramified primes, the discriminant of the maximal order or a set of integers dividing the index of $O$ in the maximal order.

MaximalOrder(O::AlgAssAbsOrd)

Given an order $O$, this function returns a maximal order containing $O$.

MaximalOrder(A::AbsAlgAss{QQFieldElem}) -> AlgAssAbsOrd

Returns a maximal order of $A$.

lll — Method

lll(M::NfAbsOrd) -> NfAbsOrd

The same order, but with the basis now being LLL reduced wrt. the Minkowski metric.

any_order — Method

any_order(K::number_field)

Return some order in $K$. In case the defining polynomial for $K$ is monic and integral, this just returns the equation order. In the other case $\mathbb Z[\alpha]\cap \mathbb Z[1/\alpha]$ is returned.

Example


julia> Qx, x = polynomial_ring(FlintQQ, "x");
julia> K, a = number_field(x^2 - 2, "a");
julia> O = EquationOrder(K)Order of Number field of degree 2 over QQ
with Z-basis NfOrdElem[1, a]

parent — Method

parent(O::NfAbsOrd) -> NfOrdSet

Returns the parent of $\mathcal O$, that is, the set of orders of the ambient number field.

signature — Method

signature(O::NumFieldOrd) -> Tuple{Int, Int}

Returns the signature of the ambient number field of $\mathcal O$.

nf — Method

nf(O::NumFieldOrd) -> NumField

Returns the ambient number field of $\mathcal O$.

basis — Method

basis(O::NfAbsOrd) -> Vector{NfAbsOrdElem}

Returns the $\mathbf Z$-basis of $\mathcal O$.

basis(I::NfAbsOrdFracIdl) -> Vector{nf_elem}

Returns the $\mathbf Z$-basis of $I$.

lll_basis — Method

lll_basis(M::NumFieldOrd) -> Vector{NumFieldElem}

A basis for $M$ that is reduced using the LLL algorithm for the Minkowski metric.

basis — Method

basis(O::NfOrd, K::AnticNumberField) -> Vector{nf_elem}

Returns the $\mathbf Z$-basis elements of $\mathcal O$ as elements of the ambient number field.

pseudo_basis — Method

  pseudo_basis(O::NfRelOrd{T, S}) -> Vector{Tuple{NumFieldElem{T}, S}}

Returns the pseudo-basis of $\mathcal O$.

basis_pmatrix — Method

  basis_pmatrix(O::NfRelOrd) -> PMat

Returns the basis pseudo-matrix of $\mathcal O$ with respect to the power basis of the ambient number field.

basis_nf — Method

  basis_nf(O::NfRelOrd) -> Vector{NumFieldElem}

Returns the elements of the pseudo-basis of $\mathcal O$ as elements of the ambient number field.

inv_coeff_ideals — Method

  inv_coeff_ideals(O::NfRelOrd{T, S}) -> Vector{S}

Returns the inverses of the coefficient ideals of the pseudo basis of $O$.

basis_matrix — Method

basis_matrix(O::NfAbsOrd) -> FakeFmpqMat

Returns the basis matrix of $\mathcal O$ with respect to the basis of the ambient number field.

basis_mat_inv — Method

basis_mat_inv(O::NfAbsOrd) -> FakeFmpqMat

Returns the inverse of the basis matrix of $\mathcal O$.

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

Return the inverse of the basis matrix of $A$.

gen_index — Method

gen_index(O::NfOrd) -> QQFieldElem

Returns the generalized index of $\mathcal O$ with respect to the equation order of the ambient number field.

is_index_divisor — Method

is_index_divisor(O::NfOrd, d::ZZRingElem) -> Bool
is_index_divisor(O::NfOrd, d::Int) -> Bool

Returns whether $d$ is a divisor of the index of $\mathcal O$. It is assumed that $\mathcal O$ contains the equation order of the ambient number field.

minkowski_matrix — Method

minkowski_matrix(O::NfAbsOrd, abs_tol::Int = 64) -> arb_mat

Returns the Minkowski matrix of $\mathcal O$. Thus if $\mathcal O$ has degree $d$, then the result is a matrix in $\operatorname{Mat}_{d\times d}(\mathbf R)$. The entries of the matrix are real balls of type arb with radius less then 2^-abs_tol.

in — Method

in(a::NumFieldElem, O::NumFieldOrd) -> Bool

Checks whether $a$ lies in $\mathcal O$.

norm_change_const — Method

norm_change_const(O::NfOrd) -> (Float64, Float64)

Returns $(c_1, c_2) \in \mathbf R_{>0}^2$ such that for all $x = \sum_{i=1}^d x_i \omega_i \in \mathcal O$ we have $T_2(x) \leq c_1 \cdot \sum_i^d x_i^2$ and $\sum_i^d x_i^2 \leq c_2 \cdot T_2(x)$, where $(\omega_i)_i$ is the $\mathbf Z$-basis of $\mathcal O$.

trace_matrix — Method

trace_matrix(O::NfAbsOrd) -> ZZMatrix

Returns the trace matrix of $\mathcal O$, that is, the matrix $(\operatorname{tr}_{K/\mathbf Q}(b_i \cdot b_j))_{1 \leq i, j \leq d}$.

+ — Method

+(R::NfOrd, S::NfOrd) -> NfOrd

Given two orders $R$, $S$ of $K$, this function returns the smallest order containing both $R$ and $S$. It is assumed that $R$, $S$ contain the ambient equation order and have coprime index.

poverorder — Method

poverorder(O::NfOrd, p::ZZRingElem) -> NfOrd
poverorder(O::NfOrd, p::Integer) -> NfOrd

This function tries to find an order that is locally larger than $\mathcal O$ at the prime $p$: If $p$ divides the index $[ \mathcal O_K : \mathcal O]$, this function will return an order $R$ such that $v_p([ \mathcal O_K : R]) < v_p([ \mathcal O_K : \mathcal O])$. Otherwise $\mathcal O$ is returned.

poverorders — Method

poverorders(O, p) -> Vector{Ord}

Returns all p-overorders of O, that is all overorders M, such that the index of O in M is a p-power.

pmaximal_overorder — Method

pmaximal_overorder(O::NfOrd, p::ZZRingElem) -> NfOrd
pmaximal_overorder(O::NfOrd, p::Integer) -> NfOrd

This function finds a $p$-maximal order $R$ containing $\mathcal O$. That is, the index $[ \mathcal O_K : R]$ is not divisible by $p$.

pradical — Method

pradical(O::NfOrd, p::{ZZRingElem|Integer}) -> NfAbsOrdIdl

Given a prime number $p$, this function returns the $p$-radical $\sqrt{p\mathcal O}$ of $\mathcal O$, which is just $\{ x \in \mathcal O \mid \exists k \in \mathbf Z_{\geq 0} \colon x^k \in p\mathcal O \}$. It is not checked that $p$ is prime.

pradical — Method

  pradical(O::NfRelOrd, P::NfOrdIdl) -> NfRelOrdIdl

Given a prime ideal $P$, this function returns the $P$-radical $\sqrt{P\mathcal O}$ of $\mathcal O$, which is just $\{ x \in \mathcal O \mid \exists k \in \mathbf Z_{\geq 0} \colon x^k \in P\mathcal O \}$. It is not checked that $P$ is prime.

ring_of_multipliers — Method

ring_of_multipliers(I::NfAbsOrdIdl) -> NfAbsOrd

Computes the order $(I : I)$, which is the set of all $x \in K$ with $xI \subseteq I$.

Invariants

discriminant — Method

discriminant(O::NfOrd) -> ZZRingElem

Returns the discriminant of $\mathcal O$.

discriminant(E::EllCrv{T}) -> T

Compute the discriminant of $E$.

discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

discriminant — Method

discriminant(O::NfOrd) -> ZZRingElem

Returns the discriminant of $\mathcal O$.

reduced_discriminant — Method

reduced_discriminant(O::NfOrd) -> ZZRingElem

Returns the reduced discriminant, that is, the largest elementary divisor of the trace matrix of $\mathcal O$.

degree — Method

degree(O::NumFieldOrd) -> Int

Returns the degree of $\mathcal O$.

index — Method

index(O::NfOrd) -> ZZRingElem

Assuming that the order $\mathcal O$ contains the equation order $\mathbf Z[\alpha]$ of the ambient number field, this function returns the index $[ \mathcal O : \mathbf Z]$.

different — Method

different(R::NfAbsOrd) -> NfAbsOrdIdl

The different ideal of $R$, that is, the ideal generated by all differents of elements in $R$. For Gorenstein orders, this is also the inverse ideal of the co-different.

codifferent — Method

codifferent(R::NfAbsOrd) -> NfOrdIdl

The codifferent ideal of $R$, i.e. the trace-dual of $R$.

is_gorenstein — Method

is_gorenstein(O::NfOrd) -> Bool

Return whether the order \mathcal{O} is Gorenstein.

is_bass — Method

is_bass(O::NfOrd) -> Bool

Return whether the order \mathcal{O} is Bass.

is_equation_order — Method

is_equation_order(O::NumFieldOrd) -> Bool

Returns whether $\mathcal O$ is the equation order of the ambient number field $K$.

zeta_log_residue — Method

zeta_log_residue(O::NfOrd, error::Float64) -> arb

Computes the residue of the zeta function of $\mathcal O$ at $1$. The output will be an element of type arb with radius less then error.

ramified_primes — Method

ramified_primes(O::NfAbsOrd) -> Vector{ZZRingElem}

Returns the list of prime numbers that divide $\operatorname{disc}(\mathcal O)$.

Arithmetic

Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are

ClassGroup
UnitGroup

All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian group, see here, for ideals, here.

For the processing of units, there are a couple of helper functions also available:

is_independent — Function

is_independent{T}(x::Vector{T})

Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.

Predicates

is_contained — Method

is_contained(R::NfAbsOrd, S::NfAbsOrd) -> Bool

Checks if $R$ is contained in $S$.

is_maximal — Method

is_maximal(R::NfAbsOrd) -> Bool

Tests if the order $R$ is maximal. This might trigger the computation of the maximal order.