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A hyperelliptic curve isomorphism is defined by a linear fractional
transformation t(x:z) = (ax + bz:cx + dz), a scale factor e, and a
polynomial u(x) of degree at most g + 1, where g is the genus of
the curve. This data defines the isomorphism of weighted projective
points
(x : y : z) |-> (ax + bz : ey + tilde(u)(x, z) : cx + dz),
where tilde(u) is the degree g + 1 homogenization of u.
When not specified, the values of e and u are by default taken
to be 1 and 0, respectively.
An isomorphism can be created from the parent structures by coercing a tuple
< [a, b, c, d], e, u > into the structure of isomorphisms between two
hyperelliptic curves, or by creating it as a transformation of a given curve,
i.e., creating the codomain curve together with the isomorphism from the given
data.
Note that due to the projective weighting of the ambient space of the
curve, two equal isomorphisms may have different representations.
Given a hyperelliptic curve C, this returns the structure of all
automorphisms of the curve.
Given hyperelliptic curves C1 and C2 of the same genus and base
field, this returns the structure of all isomorphisms between them.
Transformation(C, u) : CrvHyp, RngUPolElt -> CrvHyp, MapIsoSch
Transformation(C, e) : CrvHyp, RngElt -> CrvHyp, MapIsoSch
Transformation(C, e, u) : CrvHyp, RngElt, RngUPolElt -> CrvHyp, MapIsoSch
Transformation(C, t, e, u) : CrvHyp, [RngElt], RngElt, RngUPolElt -> CrvHyp, MapIsoSch
Returns the hyperelliptic curve C' which is the codomain of the
isomorphism from the hyperelliptic curve C
specified by the sequence of ring elements t, the ring element e and
the polynomial u, followed by the
the isomorphism to the curve.
We create the hyperelliptic curve y 2=x 5 - 7 and apply a transformation
to it.
> P<x> := PolynomialRing(Rationals());
> H1 := HyperellipticCurve(x^5-7);
> H2, phi := Transformation(H1, [0,1,1,0], 1/2, x^2+1);
> H2;
Hyperelliptic Curve defined by y^2 + (-2*x^2 - 2)*y = -7/4*x^6 - x^4 - 2*x^2 +
1/4*x - 1 over Rational Field
> phi;
(x : y : z) :-> (z : 1/2*y + x^2*z + z^3 : x)
> IsIsomorphic(H1, H2);
true (x : y : z) :-> (z : 1/2*y + x^3 + x*z^2 : x)
The "parent structure" of an isomorphism between two hyperelliptic curves
(which contains all isomorphisms with the same domain and codomain as the given
isomorphism).
The curve which is the domain of the given isomorphism.
The curve which is the target of the given isomorphism.
This section details the features for computing isomorphisms between
two curves and determining the group of automorphisms.
The genus 2 part of [LRS21] provides a function to determine the geometric
automorphism group of a genus 2 curve in any characteristic by working with
Cardona---Quer---Nart---Pujola invariants (see
Subsection Igusa Invariants). This replaces the old function that
only worked in odd (or 0) characteristics. They also provide a function that
returns a list of all possible geometric automorphism groups for genus 2 curves
over a given finite field and the number of isomorphism classes of curves with
each possible group.
The genus 3 part of [LRS21] provides the same for genus 3 hyperelliptic
curves, working with Shioda invariants. Here, the characteristic has to be 0 or greater than 7.
It is also possible to compute (geometric, reduced) isomorphisms and automorphisms by a fast direct computation in all characteristic different from 2 and all genus >1.
SetVerbose("CrvHypIso", n): Maximum: 3
This function returns true if and only if the hyperelliptic curves C1
and C2 are isomorphic as hyperelliptic curves (i.e., preserving the
involution) over their common base field. If the curves are isomorphic, an
isomorphism is returned.
Given a hyperelliptic curve C of characteristic different from 2, the
function returns a permutation group followed by an isomorphism to the group of
automorphisms of the curve over its base ring. The curve must be of genus at
least one, and the automorphism group is defined to consist of those
automorphisms which commute with the hyperelliptic involution, i.e., which
induce a well-defined automorphism of its quotient projective line. A third
return value gives the action C x G -> C.
We give an example of the computation of the automorphism group of a
genus one hyperelliptic curve.
> P<x> := PolynomialRing(GF(3));
> C1 := HyperellipticCurve(x^3+x);
> G1, m1 := AutomorphismGroup(C1);
> #G1;
> [ m1(g) : g in G1 ];
[
(x : y : z) :-> (x : y : z),
(x : y : z) :-> (x : -y : z),
(x : y : z) :-> (z : y : x),
(x : y : z) :-> (z : -y : x)
]
We note that due to the weighted projective space, the same map
may have a non-unique representation, however the equality function
is able to identify equivalence on representations.
> f := m1(G1.3);
> f;
(x : y : z) :-> (z : y : x)
> g := Inverse(f);
> g;
(x : y : z) :-> (2*z : y : 2*x)
> f eq g;
true
We see that the geometric automorphism group is much larger.
By base extending the curve to a quadratic extension, we find the
remaining automorphisms of the curve.
> K<t> := GF(3,2);
> C2 := BaseExtend(C1, K);
> G2, m2 := AutomorphismGroup(C2);
> #G2;
48
> O := C2![1,0,0];
> auts := [ m2(g) : g in G2 ];
> [ f : f in auts | O@f eq O ];
[
(x : y : z) :-> (x : y : z),
(x : y : z) :-> (x : -y : z),
(x : y : z) :-> (x : t^2*y : 2*z),
(x : y : z) :-> (x + t^6*z : t^2*y : 2*z),
(x : y : z) :-> (x + t^6*z : y : z),
(x : y : z) :-> (x + t^2*z : y : z),
(x : y : z) :-> (x + t^2*z : t^2*y : 2*z),
(x : y : z) :-> (x : t^6*y : 2*z),
(x : y : z) :-> (x + t^6*z : t^6*y : 2*z),
(x : y : z) :-> (x + t^6*z : -y : z),
(x : y : z) :-> (x + t^2*z : -y : z),
(x : y : z) :-> (x + t^2*z : t^6*y : 2*z)
]
> #$1;
12
Note that this curve is an example of a supersingular elliptic curve in
characteristic 3. In the final computation we restrict to the automorphisms
as an elliptic curve, i.e., those which fix the point at infinity --- the
identity element of the group law.
In the context of hyperelliptic curves of genus one, the group of automorphisms
must stabilize the ramification points of the hyperelliptic involution. These
are precisely the 2-torsion elements as an elliptic curve. So we have a
group extension by the 2-torsion elements, acting by translation. Converting
to an elliptic curve, we find that there are two 2-torsion elements over
F3:
> E1 := EllipticCurve(C1);
> A1 := AbelianGroup(E1);
> A1;
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0
We see that two of the 2-torsion elements are defined over F 3,
and the remaining ones appear over the quadratic extension. So in
the former case the automorphism group is an extension of the
elliptic curve automorphism group (of order 2) by Z/2Z, and in
latter case the automorphism group is an extension (of the group of
order 12) by the abelian group isomorphic to Z/2Z x Z/2Z.
Note that there exist other curve automorphisms given by translations
by other torsion points (under the addition as an elliptic curve), but
that do not commute with the hyperelliptic involution, hence do not
enter into the hyperelliptic automorphism group.
IsIsomorphicHyperellipticCurves(f1, f2) : RngUPolElt, RngUPolElt -> BoolElt, List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a boolean indicating whether a matrix T and a scalar e exist that
induce an isomorphism X1 |-> X2, or y2 = f1(x) |-> y2 =
f2(x), as well as a full list of all such pairs.
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
We determine the isomorphisms C 1 to C 2 for C 1 : y 2 = x 12 - 1 and
C 2 : y 2 = x 12 + 1. The code and the resulting output are as follows.
> P<x> := PolynomialRing(Rationals());
> C1 := HyperellipticCurve(x^12 - 1);
> C2 := HyperellipticCurve(x^12 + 1);
> test, _ := IsIsomorphicHyperellipticCurves(C1, C2);
> test;
false
> test, _ := IsIsomorphicHyperellipticCurves(C1, C2 : geometric := true);
> test;
true
IsomorphismsOfHyperellipticCurves(f1, f2) : RngUPolElt, RngUPolElt -> List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a full list of pairs of matrices T and scalars e that induce an
isomorphism X1 |-> X2, or y2 = f1(x) |-> y2 = f2(x).
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
For more details, see [LRS12].
AutomorphismsOfHyperellipticCurve(f) : RngUPolElt -> List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a full list of pairs of matrices T and scalars e that induce an
automorphism of the curve X, or the curve y2 = f(x).
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
AutomorphismGroupOfHyperellipticCurve(f, Autos) : RngUPolElt, List -> GrpPerm, Map
explicit: BoolElt Default: false
Return the automorphisms group defined by the sequence Autos, as a permutation
group (and its representation as automorphisms of the curve if explicit is set to true).
AutomorphismGroupOfHyperellipticCurve(f) : RngUPolElt -> GrpPerm, Map
geometric: BoolElt Default: false
explicit: BoolElt Default: false
Return the automorphisms group of the curve X or y2 = f(x), as a permutation
group (and its representation if explicit is set to true).
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned.
We determine automorphism group of the curve C 1 : y 2 = x 12 - 1 over the finite field F 11.
> P<x> := PolynomialRing(GF(11));
> C := HyperellipticCurve(x^12 - 1);
> aut, phi := AutomorphismGroupOfHyperellipticCurve(C : geometric := true,
> explicit := true);
> aut;
Permutation group aut acting on a set of cardinality 2640
Order = 2640 = 2^4 * 3 * 5 * 11
> GroupName(aut);
C2.PSL(2,11).C2
> [phi(aut.i) : i in [1..Ngens(aut)]];
[
<
[ 1 0]
[ 0 1],
10
>,
<
[ 1 $.1^27]
[ $.1^77 $.1^20],
4
>,
<
[ 1 $.1^98]
[ $.1^28 $.1^30],
8
>,
<
[ 1 $.1^115]
[ $.1^5 10],
$.1^6
>,
<
[ 1 $.1^85]
[ $.1^35 10],
$.1^6
>
]
GeometricAutomorphismGroupFromIgusaInvariants(GI) : SeqEnum -> SeqEnum, SeqEnum
Given a hyperelliptic curve C of genus 2 or 3 the function returns a
finitely-presented group isomorphic to the geometric automorphism group, i.e.,
the automorphism group of the curve over an algebraic closure of its base
field.
The method used for genus 2 is to compute the Cardona---Quer---Nart---Pujola
invariants (see Subsection Igusa Invariants) of C and use the
classification of the possible automorphism groups in terms of the invariants.
See [SV01] and [CQ05] for the odd (and 0) characteristic case
and [CNP05] for characteristic 2.
The method used for genus 3 is to compute the Shioda invariants (see
Subsection Shioda Invariants) of C and use the classification of
the possible automorphism groups in terms of the invariants. For this case, the
base field must have characteristic 0 or ≥11.
There is also a genus 2 version where the argument is the sequence GI of
Cardona---Quer---Nart---Pujola invariants of a curve rather than the actual curve.
This avoids actually constructing the curve, in case the user is starting from
the invariants.
The functions are part the package [LRS21] contributed by Reynald Lercier,
Christophe Ritzenthaler, and Jeroen Sijsling.
For more details, see [LRS12], [MSSV02].
There is a variant of the last intrinsic for genus 3 curves where the argument
is the sequence JI of
Shioda invariants of a curve rather than the actual
curve. This avoids actually constructing the curve, in case the user is
starting from the invariants. The same restrictions on the
characteristic of the base field apply.
We give examples of the computation of the geometric automorphism group of a
genus two and a genus three hyperelliptic curve.
> P<x> := PolynomialRing(RationalField());
> f := x^6+x^3+13;
> C := HyperellipticCurve(f);
> time GeometricAutomorphismGroup(C);
Permutation group acting on a set of cardinality 6
Order = 12 = 2^2 * 3
(1, 2, 3, 4, 5, 6)
(1, 6)(2, 5)(3, 4)
Time: 0.010
> f := x^8-1;
> C1 := HyperellipticCurve(f);
> GeometricAutomorphismGroup(C1);
GrpPC of order 32 = 2^5
PC-Relations:
$.1^2 = $.4,
$.3^2 = $.5,
$.2^$.1 = $.2 * $.3,
$.3^$.1 = $.3 * $.5,
$.3^$.2 = $.3 * $.5
Note that AutomorphismGroup can be used to retrieve the same
(and more!) information but this can be much slower.
> aut := AutomorphismGroup(C);
> aut;
Symmetric group aut acting on a set of cardinality 2
Order = 2
(1, 2)
Id(aut)
We need to extend the field!
> Qbar := AlgebraicClosure();
> Cbar := BaseChange(C, Qbar);
> time autbar := AutomorphismGroup(Cbar);
Time: 332.290
> autbar;
Permutation group autbar acting on a set of cardinality 12
Order = 12 = 2^2 * 3
(1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 12)
Id(autbar)
(1, 3)(2, 4)(5, 7)(6, 8)(9, 11)(10, 12)
(1, 5, 9)(2, 6, 10)(3, 11, 7)(4, 12, 8)
(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)(6, 12)
(1, 9, 5)(2, 10, 6)(3, 7, 11)(4, 8, 12)
(1, 11)(2, 12)(3, 5)(4, 6)(7, 9)(8, 10)
> IdentifyGroup(autbar);
<12, 4>
Given a finite field F (of any characteristic), the function returns two
sequences. The first gives a list of all possible geometric automorphism
groups for genus 2 curves defined over F and the second gives the
corresponding number of isomorphism (over bar(F), the algebraic closure of
F) classes of F-curves having the given automorphism group. The groups
are represented as finitely-presented groups.
This function belong to the genus 2 part of [LRS21] and is based on the
classification analysis in [Car03] and [CNP05].
Given a finite field F of characteristic ≥11, the function returns two
sequences. The first gives a list of all possible geometric automorphism
groups for genus 3 curves defined over F and the second gives the
corresponding number of isomorphism (over bar(F), the algebraic closure of
F) classes of F-curves having the given automorphism group. The groups
are represented as permutation groups.
This function belongs to the genus 3 part of [LRS21]. For more details,
see also [LRS12].
We determine the possible (geometric) automorphism groups for genus
2 curves over F 2.
> gps,ncls := GeometricAutomorphismGroupGenus2Classification(GF(2));
> [#gp : gp in gps];
[ 2, 12, 32, 160 ]
> ncls;
[ 5, 1, 1, 1 ]
TwistsOfHyperellipticPolynomials(fh) : SeqEnum[RngUPolElt] -> SeqEnum, GrpPerm
AutomorphismGroup: BoolElt Default: false
TwistedHyperellipticPolynomialsFromShiodaInvariants(JI: parameters) : SeqEnum[FldFinElt] -> SeqEnum, GrpPerm
RationalModel: BoolElt Default: true
When f is a polynomial defining a hyperelliptic curve C: y2=f(x) over a
finite field, these functions return the list of polynomials g such that
y2=g(x) is a twist of C (i.e. a curve over k isomorphic to C over
bar(k)). Unless the genus of C is 2 or 3, the finite field is required
to be of characteristic not 2, but in these cases it is also possible to
compute the lists of twists of y2 + h(x)y=f(x) given f and h. In all
cases, the geometric reduced automorphism groups is returned as an abstract
group.
There is also a version where the argument is JI, the Shioda invariants of a
genus 3 curve (see Section Shioda Invariants) over a finite field.
For more details, see [CN07].
TwistsFromIgusaInvariants(JI: parameters) : SeqEnum[FldFinElt] -> SeqEnum[CrvHyp], GrpPerm
TwistsFromG2Invariants(JI: parameters) : SeqEnum[FldFinElt] -> SeqEnum[CrvHyp], GrpPerm
TwistsFromShiodaInvariants(JI: parameters) : SeqEnum[FldFinElt] -> SeqEnum[CrvHyp], GrpPerm
AutomorphismGroup: BoolElt Default: false
For C a hyperelliptic curve over a finite field of characteristic not 2
(unless the genus of C is 2 or 3) or a smooth plane quartic over any
finite field, the function Twists return the list of all twists of C and the
reduced geometric automorphism group in the hyperelliptic case or the geometric
automorphism group in the smooth plane quartic case, as an abstract group.
There is also a version where the argument is JI, the sequence of Igusa or
Cardona---Quer---Nart---Pujola invariants of a genus 2 curve (see
Section Igusa Invariants) or the sequence of Shioda invariants of
a genus 3 curve (see Section Shioda Invariants) over a finite
field, which returns the full set of isomorphism classes (twists) of curves
over k with the given invariants.
AutomorphismGroup: BoolElt Default: false
Compute the twists of a curve over a finite field with respect to a subgroup
generated by geometric automorphisms (given by a list of N x N matrices) which
act on it linearly.
Computes and returns all twists of a genus 3 hyperelliptic curve and its
geometric automorphism group corresponding to a sequence of Shioda invariants
JI (see Section Shioda Invariants) over a finite field of
characteristic different from 5. In fact the first return value is a sequence
of polynomials f(x) (or of couple f, h in characteristic 2) of degree 7 or
8 such that the twisted curves correspond to y2=f(x) (or to y2 + h(x)y=f(x)
in characteristic 2). The reason for this is that JI could be a singular set of invariants corresponding to polynomials f with discriminant
zero. In that case, these do not correspond to hyperelliptic curves, but it
might be useful to get the full set of twists anyway.
We determine the twists of the hyperelliptic curve C : y 2 = x 12 - 1 over the finite field F 11.
> P<x> := PolynomialRing(GF(11));
> C := HyperellipticCurve(x^12 - 1);
> Ts, Aut := Twists(C : AutomorphismGroup := true);
> Ts;
[
Hyperelliptic Curve defined by y^2 = x^12 + 10 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 10*x^11 + 5*x + 3 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 3*x^11 + 8*x + 8 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 6*x^11 + 3*x over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 5*x^11 + 7*x + 3 over GF(11),
Hyperelliptic Curve defined by y^2 = 2*x^12 + 10*x^11 + 3*x + 6 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 7*x^11 + 5*x over GF(11),
Hyperelliptic Curve defined by y^2 = x^11 + 10*x over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 3*x^11 + 7*x + 4 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 2*x^11 + 9*x + 5 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 4*x^11 + 9*x + 5 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 4*x^11 + 5*x + 4 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 4*x^11 + 2*x + 3 over GF(11),
Hyperelliptic Curve defined by y^2 = x^12 + 7*x^11 + 7*x + 9 over GF(11)
]
> Aut;
Permutation group Aut acting on a set of cardinality 12
Order = 1320 = 2^3 * 3 * 5 * 11
(3, 4, 7, 5, 9)(6, 10, 8, 11, 12)
(3, 6, 5, 11, 4, 10, 9, 12, 7, 8)
(2, 3)(4, 5)(7, 12)(8, 10)(9, 11)
(1, 2)(4, 9)(5, 7)(6, 8)(11, 12)
> GroupName(Aut);
PSL(2,11).C2
We find all the twists of a supersingular genus 2 curve over F 2.
> P<x> := PolynomialRing(GF(2));
> C := HyperellipticCurve(x^5,P!1);
> C;
Hyperelliptic Curve defined by y^2 + y = x^5 over GF(2)
> tws,auts := Twists(C);
> tws;
[
Hyperelliptic Curve defined by y^2 + y = x^5 over GF(2),
Hyperelliptic Curve defined by y^2 + y = x^5 + x^4 over GF(2),
Hyperelliptic Curve defined by y^2 + y = x^5 + x^4 + 1 over GF(2)
]
> #auts; // auts is the geometric automorphism group of C
160
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