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(a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder
Order(K::AbsSimpleNumField, a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder

Returns the order generated by $a$. If check is set, it is checked whether $a$ 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. If cached is set, then the constructed order is cached for future use.

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# Order — Method.

Order(K::AbsSimpleNumField, A::QQMatrix; check::Bool = true) -> AbsSimpleNumFieldOrder

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.

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Order(K::AbsSimpleNumField, A::QQMatrix; check::Bool = true) -> AbsSimpleNumFieldOrder

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.

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# Order — Method.

Order(K::AbsSimpleNumField, A::ZZMatrix, check::Bool = true) -> AbsSimpleNumFieldOrder

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.

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

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

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# EquationOrder — Method.

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

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

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# MaximalOrder — Method.

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

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);

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# MaximalOrder — Method.

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

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.

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MaximalOrder(O::AlgAssAbsOrd)

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

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

Returns a maximal order of $A$.

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# lll — Method.

lll(M::AbsNumFieldOrder) -> AbsNumFieldOrder

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

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# 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.

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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 AbsSimpleNumFieldOrderElem[1, a]

# parent — Method.

parent(O::AbsNumFieldOrder) -> AbsNumFieldOrderSet

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

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# signature — Method.

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

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

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# nf — Method.

nf(O::NumFieldOrder) -> NumField

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

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# basis — Method.

basis(O::AbsNumFieldOrder) -> Vector{AbsNumFieldOrderElem}

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

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basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}

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

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# lll_basis — Method.

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

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

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# basis — Method.

basis(O::AbsSimpleNumFieldOrder, K::AbsSimpleNumField) -> Vector{AbsSimpleNumFieldElem}

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

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# pseudo_basis — Method.

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

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

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# basis_pmatrix — Method.

  basis_pmatrix(O::RelNumFieldOrder) -> PMat

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

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# basis_nf — Method.

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

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

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# inv_coeff_ideals — Method.

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

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

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# basis_matrix — Method.

basis_matrix(O::AbsNumFieldOrder) -> QQMatrix

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

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# basis_mat_inv — Method.

basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat

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

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# gen_index — Method.

gen_index(O::AbsSimpleNumFieldOrder) -> QQFieldElem

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

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# is_index_divisor — Method.

is_index_divisor(O::AbsSimpleNumFieldOrder, d::ZZRingElem) -> Bool
is_index_divisor(O::AbsSimpleNumFieldOrder, 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.

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# minkowski_matrix — Method.

minkowski_matrix(O::AbsNumFieldOrder, abs_tol::Int = 64) -> ArbMatrix

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

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# in — Method.

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

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

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# norm_change_const — Method.

norm_change_const(O::AbsSimpleNumFieldOrder) -> (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 $T_2(x)\le c_1\cdot \sum _i^d{x}_i^2$ and $\sum _i^d{x}_i^2\le c_2\cdot T_2(x)$, where $({\omega }_i{)}_i$ is the $\mathbf{Z}$-basis of $\mathcal{O}$.

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# trace_matrix — Method.

trace_matrix(O::AbsNumFieldOrder) -> ZZMatrix

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

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# + — Method.

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

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.

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# poverorder — Method.

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

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.

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# 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.

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# pmaximal_overorder — Method.

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

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

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# pradical — Method.

pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal

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

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# pradical — Method.

  pradical(O::RelNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> RelNumFieldOrderIdeal

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

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# ring_of_multipliers — Method.

ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder

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

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Invariants

# discriminant — Method.

discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem

Returns the discriminant of $\mathcal{O}$.

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# reduced_discriminant — Method.

reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem

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

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# degree — Method.

degree(O::NumFieldOrder) -> Int

Returns the degree of $\mathcal{O}$.

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# index — Method.

index(O::AbsSimpleNumFieldOrder) -> ZZRingElem

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

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# different — Method.

different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal

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.

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# codifferent — Method.

codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

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

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# is_gorenstein — Method.

is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool

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

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# is_bass — Method.

is_bass(O::AbsSimpleNumFieldOrder) -> Bool

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

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# is_equation_order — Method.

is_equation_order(O::NumFieldOrder) -> Bool

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

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# zeta_log_residue — Method.

zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem

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

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# ramified_primes — Method.

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

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

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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.

• torsion_unit_group(::AbsSimpleNumFieldOrder)

• unit_group(::AbsSimpleNumFieldOrder)

• unit_group_fac_elem(::AbsSimpleNumFieldOrder)

• sunit_group(::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}})

• sunit_group_fac_elem(::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}})

• sunit_mod_units_group_fac_elem(::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}})

• class_group(::AbsSimpleNumFieldOrder)

• picard_group(::AbsSimpleNumFieldOrder)

• narrow_class_group(::AbsSimpleNumFieldOrder)

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.

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Predicates

# is_contained — Method.

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

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

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# is_maximal — Method.

is_maximal(R::AbsNumFieldOrder) -> Bool

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

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