Quaternion algebras

We provide a model for quaternion algebras over a field $K$ in standard form, which is parametrized by two elements $a,b\in K$. The corresponding quaternion algebra has a basis $1,i,j,k$ satisfying $i^2=a$, $j^2=b$ and $ij=-ji=k$.

This functionality is currently restricted to fields of characteristic not equal to two.

Creation

quaternion_algebra Method

quaternion_algebra(K::Field, a, b) -> QuaternionAlgebra

If the characteristic of K isn't 2, return the quaternion algebra $(a,b{)}_K$ defined by $i^2=a$, $j^2=b$, and $ij=-ji$.

If the characteristic of K is 2, return the quaternion algebra $[a,b{)}_K$ defined by $i^2+i=a$, $j^2=b$, and $ij=ji+j$.

Examples

julia> Q = quaternion_algebra(QQ, -1, -1)
Quaternion algebra
  over rational field
  defined by i^2 = -1, j^2 = -1, ij = -ji

julia> K, sqrt2 = quadratic_field(2);

julia> Q = quaternion_algebra(K, sqrt2, -1)
Quaternion algebra
  over real quadratic field defined by x^2 - 2
  defined by i^2 = sqrt(2), j^2 = -1, ij = -ji

julia> R, x = polynomial_ring(GF(2), "x"); F=fraction_field(R);

julia> Q = quaternion_algebra(F, 0, x)
Quaternion algebra
  over fraction field of R
  defined by i^2 = i+0, j^2 = x, ij = ji+j

source

quaternion_algebra_ramified_at Method

quaternion_algebra_ramified_at(K::QQField, p::IntegerUnion;
                               check::Bool = true)
  -> QuaternionAlgebra{QQFieldElem}

Return the quaternion algebra over $\mathbf{Q}$ that is ramified exactly at the prime $p$ and at infinity, presented as $(-p,-q{)}_{\mathbf{Q}}$, so $i^2=-p$ and $j^2=-q$, where $q$ is the least prime with $q\equiv 3mod8$ and $\left(\frac{p}{q}\right)=-1$.

The algebra carries the maximal order, which maximal_order returns,

$$\mathbf{Z}+\mathbf{Z}\frac{1+j}{2}+\mathbf{Z}\frac{i(1+j)}{2}+\mathbf{Z}\frac{(r+i)j}{q},$$

where $r$ is the square root of $-p$ modulo $q$ in $(0,q/2]$. Both $q$ and $r$ are determined by $p$, so the algebra and its maximal order are the same on every call.

If check is true, it is tested that $p$ is prime.

The presentation and the order are those of [1], Lemmas 1.1 and 1.2, with the least admissible $q$, which exists by Dirichlet's theorem.

Examples

julia> A = quaternion_algebra_ramified_at(QQ, 97)
Quaternion algebra
  over rational field
  defined by i^2 = -97, j^2 = -19, ij = -ji

julia> abs(discriminant(maximal_order(A))) == 97^2
true

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Arithmetic of elements

conj Method

conj(a::AssociativeAlgebraElem{_, QuaternionAlgebra})
                             -> AssociativeAlgebraElem{_, QuaternionAlgebra}

Return the image of $a$ under the canonical involution of the quaternion algebra.

Examples

julia> Q = quaternion_algebra(QQ, -1, -1); a = Q([1, 1, 1, 1])
1 + i + j + k

julia> conj(a)
1 - i - j - k

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trred Method

trred(x::AbstractAssociativeAlgebraElem{T}) where T -> T

Returns the reduced trace of $x$.

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normred Method

normred(x::AbstractAssociativeAlgebraElem{T}) where T -> T

Returns the reduced norm of $x$.

source

reduced_charpoly Method

reduced_charpoly(a::AbstractAssociativeAlgebraElem) -> PolyRingElem

Returns the reduced characteristic polynomial of $a$ as a polynomial over base_ring(algebra(a)).

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Example

julia> Q = quaternion_algebra(QQ, -1, -1)
Quaternion algebra
  over rational field
  defined by i^2 = -1, j^2 = -1, ij = -ji

julia> z = Q([1, 2, 0, 1//3])
1 + 2*i + 1//3*k

julia> trred(z)
2

julia> normred(z)
46//9

julia> reduced_charpoly(z)
x^2 - 2*x + 46//9

julia> m = reduced_charpoly(z)
x^2 - 2*x + 46//9

julia> m(z)
0

Eichler orders

is_eichler Method

is_eichler(O::Union{AlgAssAbsOrd, AlgAssRelOrd}) -> Bool

Return whether the order $O$ of a quaternion algebra is an Eichler order, that is, the intersection of two maximal orders.

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eichler_level Method

eichler_level(O::AlgAssAbsOrd{<:QuaternionAlgebra}) -> ZZRingElem
eichler_level(O::AlgAssRelOrd{<:Any, <:Any, <:QuaternionAlgebra})
  -> AbsNumFieldOrderIdeal

Return the level of the Eichler order $O$ of a quaternion algebra $A$ over $K$: the ideal $M$ of the ring of integers of $K$, coprime to the discriminant $D$ of $A$, with $\mathrm{discrd}(O)=DM$ [2], 23.4.19. Over $\mathbf{Q}$ it is returned as a positive integer.

An ArgumentError is thrown if $O$ is not an Eichler order, see is_eichler.

Examples

The order of the matrices $\left(\begin{array}{cc}a& b\\ 6c& d\end{array}\right)$, with $a,b,c,d\in \mathbf{Z}$, in $M_2(\mathbf{Q})\cong (1,1{)}_{\mathbf{Q}}$, spanned by the matrix units $(1\pm i)/2$, $(j+k)/2$ and $6$ times $(j-k)/2$:

julia> A = quaternion_algebra(QQ, 1, 1);

julia> e, i, j, k = basis(A);

julia> h = QQ(1, 2);

julia> O = order(A, [h * (e + i), h * (e - i), h * (j + k), 3 * (j - k)]);

julia> eichler_level(O)
6

julia> eichler_level(maximal_order(A))
1

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Right ideal classes and types

right_class_set Method

right_class_set(O::Union{AlgAssAbsOrd, AlgAssRelOrd}) -> Vector

Return representatives of the classes of locally principal right ideals of the order $O$ of a quaternion algebra $A$ over $\mathbf{Q}$ or a number field. Two ideals $I$ and $J$ lie in the same class if $J=\alpha I$ for some $\alpha \in A^{\times }$.

The representatives are integral right ideals of $O$.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> C = right_class_set(O);

julia> length(C), all(I -> right_order(I) == O, C)
(8, true)

julia> length(right_class_set(maximal_order(quaternion_algebra(QQ, -1, 3))))
1

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type_set Method

type_set(O::Union{AlgAssAbsOrd, AlgAssRelOrd}) -> Vector

Return representatives of the types of the order $O$ of a quaternion algebra $A$ over $\mathbf{Q}$ or a number field: one order for each isomorphism class of orders locally isomorphic to $O$.

If $A$ is not totally definite, $O$ must be an Eichler order.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> length(type_set(O)), length(right_class_set(O))
(5, 8)

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type_number Method

type_number(O::Union{AlgAssAbsOrd, AlgAssRelOrd}) -> Int

Return the number of types of the order $O$, that is, the length of type_set.

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Splitting of quaternion algebras

is_split_with_zero_divisor Function

is_split_with_zero_divisor(A::QuaternionAlgebra) -> Bool, AssociativeAlgebraElem

Given a quaternion algebra $A$, return whether $A$ is split together with an element, which is a zero-divisor in case $A$ is split.

Examples

julia> A = quaternion_algebra(QQ, 1, 4);

julia> is_split_with_zero_divisor(A)
(true, 1 + i)

julia> A = quaternion_algebra(QQ, -1, -1);

julia> is_split_with_zero_divisor(A)
(false, 0)

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Gross lattices

gross_lattice Method

gross_lattice(O::AlgAssAbsOrd{<:QuaternionAlgebra{QQFieldElem}}) -> ZZLat

Return the Gross lattice of an order $O$ of $B_p$, the quaternion algebra over $\mathbf{Q}$ ramified exactly at a prime $p$ and infinity: the image of $O$ under $x\mapsto 2x-\mathrm{trd}(x)$, which is the set of elements of reduced trace zero in $\mathbf{Z}+2O$ [3, 4]. It is a positive definite lattice of rank $3$, with the reduced norm as its quadratic form. The order need not be maximal; an ArgumentError is thrown if the algebra of $O$ is not of the form $B_p$.

It lies in the quadratic space on the coordinates with respect to basis(A) whose bilinear form is $(x,y)\mapsto \mathrm{trd}(x\overline{y})/2$, so that the norm of a vector is the reduced norm of the element and not twice it. Its basis consists of the coordinates of gross_lattice_basis and its Gram matrix is gross_gram_matrix of that basis.

For a maximal order of $B_p$, the determinant of the Gram matrix is $4p^2$ [3], and two maximal orders of $B_p$ are isomorphic if and only if their Gross lattices are isometric [4].

Examples

julia> A = quaternion_algebra(QQ, -1, -3);

julia> L = gross_lattice(maximal_order(A));

julia> rank(L), det(L)
(3, 36)

source

gross_lattice_basis Method

gross_lattice_basis(O::AlgAssAbsOrd{<:QuaternionAlgebra{QQFieldElem}})
  -> Vector{AssociativeAlgebraElem{QQFieldElem}}

Return a basis of the Gross lattice of an order $O$ of $B_p$, the elements of reduced trace zero in $\mathbf{Z}+2O$, as three elements of the algebra of $O$; see gross_lattice.

Examples

julia> A = quaternion_algebra(QQ, -1, -3);

julia> B = gross_lattice_basis(maximal_order(A));

julia> length(B), all(iszero, trred.(B))
(3, true)

source

gross_gram_matrix Method

gross_gram_matrix(B::Vector{<:AssociativeAlgebraElem{QQFieldElem}})
  -> ZZMatrix

Return the Gram matrix of the three elements in B, a basis of a Gross lattice of an order of $B_p$ (see gross_lattice), with respect to the bilinear form $(x,y)\mapsto \mathrm{trd}(x\overline{y})/2$: the reduced norms of the elements are on its diagonal. The matrix is integral for a basis of a Gross lattice; an ArgumentError is thrown otherwise.

Examples

julia> A = quaternion_algebra(QQ, -1, -3);

julia> B = gross_lattice_basis(maximal_order(A));

julia> G = gross_gram_matrix(B);

julia> det(G), [G[i, i] for i in 1:3] == normred.(B)
(36, true)

source

Ideals connecting maximal orders

connecting_ideal Method

connecting_ideal(O1::AlgAssAbsOrd{<:QuaternionAlgebra},
                 O2::AlgAssAbsOrd{<:QuaternionAlgebra}; check::Bool = true)
  -> AlgAssAbsOrdIdl
connecting_ideal(O1::AlgAssRelOrd{<:Any, <:Any, <:QuaternionAlgebra},
                 O2::AlgAssRelOrd{<:Any, <:Any, <:QuaternionAlgebra};
                 check::Bool = true) -> AlgAssRelOrdIdl

Given maximal orders $O_1$ and $O_2$ of a quaternion algebra over $\mathbf{Q}$ or a number field, return the lattice $O_1{O}_2$ spanned by the products $xy$ with $x\in O_1$ and $y\in O_2$. It is an invertible fractional ideal with left order $O_1$ and right order $O_2$ [2], Lemma 17.4.7. As it contains $O_1$ and $O_2$, it is integral only if $O_1=O_2$.

If check is true, an ArgumentError is thrown unless both orders are maximal.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> O1, O2 = left_order.(right_class_set(O)[1:2]);

julia> J = connecting_ideal(O1, O2);

julia> left_order(J) == O1, right_order(J) == O2
(true, true)

source

least_norm_ideal Method

least_norm_ideal(I::AlgAssAbsOrdIdl, O::AlgAssAbsOrd; side::Symbol = :right,
                 check::Bool = true) -> ZZRingElem, AlgAssAbsOrdIdl

Given an invertible fractional ideal $I$ with right order $O$ (side = :right) or left order $O$ (side = :left), in a definite quaternion algebra $A$ over $\mathbf{Q}$, return the least reduced norm $d$ of an integral ideal in the class of $I$, and one such ideal $J$. The class of $I$ consists of the ideals $\alpha I$ (side = :right) or $I\alpha$ (side = :left) with $\alpha \in A^{\times }$, as in right_class_set: the order $O$ of $J$ on that side is $O$ itself, while its order on the other side is a conjugate of the one of $I$. In particular, $d=1$ if and only if $I$ is principal.

For side = :right, the integral ideals in the class of $I$ are the $\overline{\gamma }I/\mathrm{nrd}(I)$ with $0\ne \gamma \in I$, of reduced norm $\mathrm{nrd}(\gamma )/\mathrm{nrd}(I)$ ([2], proof of Proposition 17.5.6), and similarly for side = :left. So $d$ is the minimum of $\mathrm{nrd}/\mathrm{nrd}(I)$ on $I$, found by one shortest vector computation, and $J$ is the ideal of such a $\gamma$.

If check is true, an ArgumentError is thrown unless $I$ is invertible with the order $O$ on the given side.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> C = right_class_set(O);

julia> d, J = least_norm_ideal(C[2], O);

julia> d == normred(J, O), right_order(J) == O, is_subset(J, O)
(true, true, true)

julia> first(least_norm_ideal(3 * C[2], O)) == d
true

source

least_connecting_norm Method

least_connecting_norm(O1::AlgAssAbsOrd, O2::AlgAssAbsOrd;
                      check::Bool = true) -> ZZRingElem, AlgAssAbsOrdIdl

Given maximal orders $O_1$ and $O_2$ of a definite quaternion algebra $A$ over $\mathbf{Q}$, return the least reduced norm $d$ of an integral ideal with left order $O_1$ and right order conjugate to $O_2$, and one such ideal. So $O_1$ is fixed and $O_2$ is taken up to conjugacy: $d$ depends only on the types of $O_1$ and $O_2$ (see type_set), does not change when they are swapped, and is $1$ if and only if they are conjugate. When $A$ is ramified only at a prime $p$ and $\mathrm{\infty }$, and $E_1$, $E_2$ are supersingular elliptic curves over ${\bar{\mathbf{F}}}_p$ with endomorphism rings $O_1$ and $O_2$, it is the least degree of an isogeny from $E_1$ to $E_2$ or to its Frobenius conjugate $E_2^{(p)}$ ([2], Proposition 42.2.16 and Corollary 42.2.21; the ideals $J$ and $JP$ below). The curve $E_2^{(p)}$ also has endomorphism ring $O_2$, and is not isomorphic to $E_2$ unless $j(E_2)\in {\mathbf{F}}_p$ ([2], Lemma 42.4.1), so $d$ need not be the least degree of an isogeny $E_1\to E_2$: for $O_1=O_2$ it is $1$, while in $B_{97}$ some $E_2\to E_2^{(p)}$ have least degree $2$.

The ideals with left order $O_1$ and right order conjugate to $O_2$ lie in the left classes of $JP$, where $J$ is connecting_ideal(O1, O2) and $P$ runs over the products of the two-sided prime ideals of $O_2$ above distinct primes ramified in $A$ ([2], Theorem 18.1.3 and Proposition 18.5.10). So $d$ is the least of the values of least_norm_ideal with side = :left on these $2^r$ ideals, where $r$ is the number of ramified primes.

If check is true, an ArgumentError is thrown unless both orders are maximal.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> Ts = type_set(O);

julia> [first(least_connecting_norm(Ts[1], T)) for T in Ts]
5-element Vector{ZZRingElem}:
 1
 2
 2
 4
 3

source

denominator Method

denominator(R::AlgAssAbsOrd, O::AlgAssAbsOrd) -> ZZRingElem

Return the least positive integer $d$ with $dR\subseteq O$, for orders $R$ and $O$ of the same algebra.

Examples

julia> O = maximal_order(quaternion_algebra_ramified_at(QQ, 97));

julia> O1, O2 = left_order.(right_class_set(O)[1:2]);

julia> denominator(O1, O1), denominator(O2, O1), index(intersect(O1, O2), O1)
(1, 2, 2)

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