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Subsections
The univariate polynomials f(x), h(x), in that order, defining the hyperelliptic
curve C by y2 + h(x)y = f(x).
Degree(C) : SetPtHyp -> RngIntElt
The degree of the hyperelliptic curve C or a pointset C of a
hyperelliptic curve.
The discriminant of the hyperelliptic curve C.
The genus of the hyperelliptic curve C.
The Clebsch, Igusa--Clebsch and Igusa invariants may be computed for curves
of genus 2.
The Magma package implementing the functions was written by Everett W. Howe
(however@alumni.caltech.edu) with some advice from Michael Stoll and is based
on some gp routines written by Fernando Rodriguez--Villegas as part of
the Computational Number Theory project funded by a TARP grant.
The gp
routines may be found at
http://www.ma.utexas.edu/users/villegas/cnt/inv.gp.
In addition, a package of functions wriiten by Reynald Lercier and Christophe
Ritzenthaler has been added which contains, amongst other things, functionality
for working with a different set of absolute (as opposed to weighted projective)
invariants referred to as Cardona-Quer-Nart-Pujola invariants. These work in
characteristic 2 as well as other characteristics, although the definitions
are different in the two cases.
Rodriguez--Villegas's routines are based on a paper of Mestre [Mes91].
The first part of Mestre's paper summarizes work of Clebsch and Igusa, and is
based on the classical theory of invariants.
This package contains functions to compute three types of invariants of
quintic and sextic polynomials f (or, perhaps more accurately, of binary
sextic forms):
 - The Clebsch invariants A, B, C, D of f,
as defined on p. 317 of Mestre;
- The Igusa--Clebsch invariants A', B', C', D'
of f, as defined on p. 319 of Mestre; and
- The Igusa invariants (or Igusa J-invariants, or
J-invariants) J2, J4, J6, J8, J10 of f, as defined
on p. 324 of Mestre.
The corresponding functions are ClebschInvariants,
IgusaClebschInvariants, and JInvariants, respectively.
For convenience, we use IgusaInvariants as a synonym for
JInvariants.
Igusa invariants may be defined for a curve of genus 2 over any field
and for polynomials of degree at most 6 over fields of characteristic
not equal to 2. The Igusa invariants of the curve y2 + hy = f are
equal to the Igusa invariants of the polynomial h2 + 4 * f except
in characteristic 2, where the latter are not defined. In practice,
the functions below will not calculate the Igusa invariants of a polynomial
unless 2 is a unit in the coefficient ring. However, Igusa invariants of
curves are available for all coefficient rings. (But see below.)
Igusa invariants are given by a sequence [ J2, J4, J6, J8, J10 ] of
five elements of the coefficient ring of the polynomials defining the curve.
This sequence should be thought of as living in weighted projective
space, with weights 2, 4, 6, 8, and 10.
It should be noted that many of the people who work with genus 2 curves over
the complex numbers prefer not to work with the real Igusa invariants, but
rather work with some related numbers, [ I2, I4, I6, I10 ]
(or [ A', B', C', D' ] in Mestre's terminology), that we call
the Igusa--Clebsch invariants, of the curve.
Once again, these live in
weighted projective space. The Igusa--Clebsch invariants of a polynomial
are defined in terms of certain nice symmetric polynomials in its roots,
and, in characteristic zero, the J-invariants may be obtained from the
I-invariants by some simple homogeneous transformations. In fact, many
of the genus-2-curves-over-the-complex-numbers people refer to the
elements i1 := I25/I10, i2 := I23 *
I4/I10
and i3 := I22 * I6/I10
as the "invariants" of the curve. The problem with the Igusa--Clebsch
invariants is that they do not work in characteristic 2 and it was for
this reason that Igusa defined his J-invariants.
The coefficient ring of the polynomial f must be an algebra over a field of
characteristic not equal to 2 or 3.
The Cardona-Quer-Nart-Pujola invariants are three absolute invariants
g1, g2, g3 which can be derived from the J-invariants and which provide
an affine classification of all genus two curves over a basefield k up to
isomorphism over bar(k). That is, there is a 1-1 correspondence between
bar(k)-isomorphism classes of such curves and triples (g1, g2, g3) in
k3. There are also functions to construct a curve with given invariants
and to find all twists of such a curve (ie representatives of the
k-isomorphism classes in the given bar(k)-isomorphism class), which will
be described in later sections.
The invariants are different in the characteristic 2 and odd (or 0) characteristic
cases. Details about the former case may be found in [CNP05]. See
[CQ05] for the latter case. In the odd characteristic case, the formulae
for [g1, g2, g3] in terms of the J-invariants are as follows:
[((J25)/(J10)), ((J23J4)/(J10)), ((J22J6)/(J10))]
J2 ≠0
[0, ((J45)/(J210)), ((J4J6)/(J10))] J2 = 0, J4 ≠0
[0, 0, ((J65)/(J210))] J2 = J4 = 0
In the characteristic 2 case, the field k must be perfect. Formulae for
the invariants (labelled ji rather than gi) may be found on
p. 191 of [CNP05].
Given a hyperelliptic curve C having genus 2, compute the Clebsch
invariants A, B, C and D as described on p. 317 of [Mes91].
The base field of C may not have characteristic 2, 3 or 5.
The invariants are found using Überschiebungen.
Given a polynomial f of degree at most 6, compute the Clebsch
invariants A, B, C and D as described on p. 317 of [Mes91].
The coefficient ring of the polynomial f must be an algebra over a field of
characteristic not equal to 2, 3 or 5. The invariants are found using
Überschiebungen.
Quick: BoolElt Default: false
Given a curve C of genus 2 defined over a field,
the Igusa--Clebsch invariants A', B', C' and D'
as described on p. 319 of [Mes91] are
found. These will be all be zero in characteristic 2.
If Quick is true, the base field of C may not have
characteristic 2, 3 or 5 and a faster method using Überschiebungen
is employed; otherwise, universal formulae are used.
Given a polynomial h having degree at most 3 and a polynomial f
having degree at most 6, the Igusa--Clebsch invariants A', B', C' and D'
of the curve y2 + hy - f = 0 are found. These will be all be zero
in characteristic 2.
Quick: BoolElt Default: false
Given a polynomial f having degree at most 6 and defined over a ring
in which 2 is a unit, the Igusa--Clebsch invariants A', B', C' and D'
of the polynomial f are found.
These will be all be zero in characteristic
2. If Quick is true, the coefficient ring of f may not have
characteristic 2, 3 or 5 and a faster method using Überschiebungen
is employed; otherwise, universal formulae are used.
JInvariants(C: parameters): CrvHyp -> SeqEnum
Quick: BoolElt Default: false
Given a curve C of genus 2 defined over a field, the function
returns the Igusa invariants (or J-invariants)
J2, J4, J6, J8, J10 as described
on p. 324 of [Mes91].
If Quick is true, the base field of C may not have
characteristic 2, 3 or 5 and a faster method using Überschiebungen
is employed; otherwise, universal formulae are used.
JInvariants(f, h): RngUPolElt, RngUPolElt -> SeqEnum
Given a polynomial h having degree at most 3 and a polynomial f
having degree at most 6, this function returns the Igusa invariants
(or J-invariants)
J2, J4, J6, J8, J10 of the curve y2 + hy = f. The
coefficient ring R of the polynomials h and f must either (a)
have characteristic 2, or (b) be a ring in which Magma can apply the
operator ExactQuotient(n,2). For example, R may be an
arbitrary field, the ring of rational integers, a polynomial ring over
a field or over the integers and so forth. However, R may not be a
p-adic ring, for instance. If the desired coefficient ring does not
meet either condition (a) or condition (b), then ScaledIgusaInvariants should be used and its invariants then scaled by
the appropriate powers of 1/2.
JInvariants(f: parameters) : RngUPolElt -> SeqEnum
Quick: BoolElt Default: false
Given a polynomial f having degree at most 6 which is defined
over a ring in which 2 is a unit, return the Igusa invariants
(or J-invariants) J2, J4, J6, J8, J10 of f.
If Quick is true, the coefficient ring of f may not have
characteristic 2, 3 or 5 and a faster method using Überschiebungen
is employed; otherwise, universal formulae are used.
Given a polynomial h having degree at most 3 and a polynomial f
having degree at most 6, return the Igusa J-invariants
of the curve y2 + hy = f, scaled by
[16, 162, 163, 164, 165].
Given a polynomial f having degree at most 6 which is defined
over a ring not of characteristic 2, return the Igusa J-invariants
of f, scaled by
[16, 162, 163, 164, 165].
Given a curve C of genus 2 defined over a field, the function
computes the ten absolute invariants of C as described on p. 325
of [Mes91].
Convert Clebsch invariants in the sequence Q to Igusa--Clebsch invariants.
Convert Igusa--Clebsch invariants in the sequence S to Clebsch invariants.
Compute and return the sequence of three Cardona-Quer-Nart-Pujola
invariants (see the introduction above) for C of genus 2.
Convert the sequence of Cardona-Quer-Nart-Pujola invariants
(see the introduction above) to a
corresponding sequence of Igusa J-invariants.
Convert the sequence of Igusa J-invariants to Cardona-Quer-Nart-Pujola
invariants (see the introduction above).
BaseRing(C) : Sch -> Fld
CoefficientRing(C) : Sch -> Fld
The base field of the hyperelliptic curve C.
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