# Isomorphisms ## Isomorphisms of Algebras Two quaternion algebras $A, B$ over a common field $F$ are isomorphic algebras if and only if they share the same ramified places. Finding an explicit isomorphism is much harder. Currently Magma embeds the first standard generator of $A$ into $B$ and then finds another element perpendicular to that image having the correct minimal polynomial. In particular, this requires the construction of two points on a conic (or equivalently, the solution of two norm equations) over quadratic extensions of $F$. ### `IsIsomorphic(A, B): AlgQuat, AlgQuat -> BoolElt, Map` ```magma Isomorphism: BoolElt Default: false ``` Given two quaternion algebras $A,B$ over ${\mathbb{Q}}$, ${\mathbb{F}}_q(X)$ with $q$ odd or a number field, this function returns `true` if and only if they are isomorphic. If the algebras are isomorphic and `Isomorphism` is set to true, an isomorphism $A \to B$ is also returned. ## Isomorphisms of Orders Two orders $S, T$ in a quaternion algebra $A$ are isomorphic if and only if they are conjugate in $A$. In a definite algebra, we use the fact that this conjugation induces an isometry of the quadratic ${\mathbb{Z}}$- or ${\mathbb{F}}_q[X]$-modules $S$ and $T$ equipped with the (absolute) norm form. Over an indefinite algebra, we use the fact that any connecting ideal $I$ having left order $S$ and right order $T$ must be isomorphic to a right ideal of $T$. See [[Kirschmer and Voight, 2010](../../references.md#cite-kirschmer-voight)] for details. ### `IsIsomorphic(S, T): AlgQuatOrd, AlgQuatOrd -> BoolElt, Map, AlgQuatElt` ### `IsIsomorphic(S, T): AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] -> BoolElt, Map, AlgQuatElt` ### `IsConjugate(S, T): AlgAssVOrd, AlgAssVOrd -> BoolElt, Map, AlgQuatElt` ```magma FindElement : BoolElt Default: false ConnectingIdeal: AlgAssVOrdIdl Default: ``` Given orders $S$ and $T$ in a quaternion algebra $A$ over ${\mathbb{Q}}$, ${\mathbb{F}}_q(X)$ (with $q$ odd) or a number field, this function returns `true` if and only if $S$ and $T$ are isomorphic. For indefinite algebras, the orders are currently required to be maximal. If `FindElement` is set, the second return value is an isomorphism from $S$ to $T$ and the third return value is an element $a$ with $T = a^{-1} S a$, inducing the isomorphism. For indefinite algebras, this search is expensive and may sometimes fail (as in `IsIsomorphic` for ideals, see below). For indefinite algebras defined over number fields, part of the computation may be faster when a connecting ideal is specified using the parameter `ConnectingIdeal`; this should be an ideal with left order $S$ and right order $T$. ### `Isomorphism(S, T): AlgQuatOrd, AlgQuatOrd -> Map` Given two isomorphic definite quaternion orders $S, T$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$ (with $q$ odd), this function returns an algebra isomorphism. For orders over number rings the intrinsic `IsIsomorphic` should be used with the optional argument `FindElement` set. ## Isomorphisms of Ideals Two right (left) ideals $I, J$ of an order $O$ in a quaternion algebra $A$ are isomorphic $O$-modules if and only if $xI=J$ ($Ix=J$) for some $x \in A^*$. To decide whether $I$ and $J$ are isomorphic, we test whether the *colon ideal* $(J:I) = \{x \in A: xI \subset J\}$ (similarly defined if $I, J$ are left ideals) is principal, or not. Over ${\mathbb{Z}}$, to accomplish this task we compute a Minkowski-reduced Gram matrix of $(J:I)$ which produces an element of smallest norm. Over ${\mathbb{F}}_q[X]$ we compute a Gram matrix which has dominant diagonal form (see [[Gerstein, 2003](../../references.md#cite-forms-fqx)]) which also produces a suitable element. Over other number rings $R$, we search $(J:I)$ for an element of the required norm by computing a reduced basis. This method runs very quickly for “reasonably small” input. (See [[Dembele and Donnelly, 2008](../../references.md#cite-dembele-donnelly), [Kirschmer and Voight, 2010](../../references.md#cite-kirschmer-voight)] for further details.) ### `IsIsomorphic(I, J): AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt, AlgAssVElt` Given ideals $I$ and $J$ of the same order $O$ in a quaternion algebra $A$ over ${\mathbb{Q}}$, ${\mathbb{F}}_q(X)$ (with $q$ odd) or a number field, this function returns `true` if and only if the quaternion ideals $I$ and $J$ are isomorphic (left or right) $O$-ideals. If $I$ and $J$ are isomorphic, there exists some $x\in A$ such that $xI=J$ (in the case of right $O$-ideals) or $Ix=J$ (for left ideals). For definite algebras, such an element $x$ is always returned. For indefinite algebras over ${\mathbb{Z}}$ or a number ring, a search for such an element is made, and if one is found then it is returned. ### `IsPrincipal(I): AlgAssVOrdIdl -> BoolElt, AlgQuatElt` Given a left (or right) ideal $I$ over ${\mathbb{Z}}, {\mathbb{F}}_q[X]$ or a number ring, this function returns `true` if and only if $I$ is a principal ideal; if so, a generator is returned as the second value. ### `IsLeftIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl -> BoolElt, Map, AlgQuatElt` ### `IsRightIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl -> BoolElt, Map, AlgQuatElt` Given two definite ideals over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$ (with $q$ odd) with the same left (or right) order $S$, this function returns `true` if and only if they are isomorphic as $S$-modules. The isomorphism and the transforming scalar in the quaternion algebra are returned as second and third values if `true`. ### `IsLeftIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> BoolElt, AlgQuatElt` ### `IsRightIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> BoolElt, AlgQuatElt` Given two left (or right) ideals $I$ and $J$ over a number ring, with the same left order $O$, this function returns `true` if and only if they are isomorphic as $O$-modules. The isomorphism is given by multiplication as a second return value. ### `LeftIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl -> Map, AlgQuatElt` Given two isomorphic left ideals over a definite order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$, this function returns the $S$-module isomorphism between them, followed by the quaternion algebra element which defines the isomorphism by right-multiplication. ### `RightIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl -> Map, AlgQuatElt` Given two isomorphic right ideals over a definite order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$, this function returns the $S$-module isomorphism between them, followed by the quaternion algebra element which defines the isomorphism by left-multiplication. ## Examples ### `Example: Isomorphism Algebras (ex-f5c0d5)` In this example, we create two quaternion algebras over ${\mathbb{F}}_7$, show that they are isomorphic and find an isomorphism between them. ```magma > F := RationalFunctionField( GF(7) ); > Q1 := QuaternionAlgebra< F | (x^2+x-1)*(x+1), x >; > a := x^3 + x^2 + 3; > b := x^13 + 4*x^11 + 2*x^10 + x^9 + 6*x^8 + 4*x^5 + 3*x^4 + x; > Q2:= QuaternionAlgebra< F | a, b >; > ok, phi:= IsIsomorphic(Q1, Q2 : Isomorphism); > ok; true > forall{ : x,y in Basis(Q1) | phi(x*y) eq phi(x)*phi(y) }; true ``` ### `Example: Isomorphism Example (ex-50db58)` In this example, we create two ideals, show that they have isomorphic right orders, and then explicitly exhibit the isomorphism. ```magma > A := QuaternionAlgebra(37); > S := MaximalOrder(A); > ideals := LeftIdealClasses(S); > _, I, J := Explode(ideals); > R := RightOrder(I); > Q := RightOrder(J); > IsIsomorphic(R,Q); true ``` Now we find an element $pi$ which conjugates R to Q, and then check that it has this property. ```magma > _, pi := Isomorphism(R,Q); > J := lideal< S | [ x*pi : x in Basis(J) ] >; > RightOrder(J) eq R; true ``` ### `Example: Left Right Isomorphisms (ex-03d026)` We construct two non-isomorphic left ideals with the same left and right orders, then investigate their isomorphisms as right ideals. ```magma > S := QuaternionOrder(37); > ideals := LeftIdealClasses(S); > _, I, J := Explode(ideals); > R := RightOrder(I); > _, pi := Isomorphism(R,RightOrder(J)); > J := lideal< S | [ x*pi : x in Basis(J) ] >; > IsLeftIsomorphic(I,J); false > IsRightIsomorphic(I,J); true Mapping from: AlgQuatOrd: I to AlgQuatOrd: J given by a rule [no inverse] 1 + i - 2*k > h, x := RightIsomorphism(I,J); > y := [1,2,-1,3]; > y := &+[y[i]*b[i] : i in [1 .. 4]] where b is Basis(I); > h(y); [-73 15 31 4] > x*y; -73 + 15*i + 31*j + 4*k ``` The existence of an isomorphism as a right ideal is due to the fact that the two-sided ideals of $R$ do not have non-isomorphic counterparts in S. ```magma > TwoSidedIdealClasses(R); [ Ideal with basis Pseudo-matrix over Integer Ring 1 * [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] , Ideal with basis Pseudo-matrix over Integer Ring 1 * [37 0 32 18] [ 0 37 10 2] [ 0 0 1 0] [ 0 0 0 1] ] > TwoSidedIdealClasses(S); [ Ideal with basis Pseudo-matrix over Integer Ring 1 * [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] ] ``` Thus while `Conjugate(I)*J` is in the non-principal $R$-ideal class, the ideal `I*Conjugate(J)` represents the unique principal ideal class of $S$. ### `Example: Left Right Isomorphisms Number Field (ex-0c04cb)` We exhibit isomorphism testing for ideals of orders over number rings. ```magma > P := PolynomialRing(Rationals()); > F := NumberField(x^3-3*x-1); > Z_F := MaximalOrder(F); > F := FieldOfFractions(Z_F); > A := QuaternionAlgebra; > O := Order([alpha,beta,alphabeta]); > O; Order of Quaternion Algebra with base ring F with coefficient ring Maximal Equation Order with defining polynomial x^3 - 3*x - 1 over its ground order > I := ideal; > I eq (I + ideal); true > I eq (I + ideal); false > > Foo := InfinitePlaces(F); > A := QuaternionAlgebra(ideal, Foo); > IsDefinite(A); true > O := MaximalOrder(A); > I := rideal; > J := rideal; > IsIsomorphic(I, J); true (F.2 + F.3) + (27/9190*F.1 - 143/9190*F.2 - 73/9190*F.3)*i + (-251/27570*F.1 + 7/2757*F.2 + 10/2757*F.3)*k ```