Mestre’s algorithm#
This file contains functions that:
create hyperelliptic curves from the Igusa-Clebsch invariants (over \(\QQ\) and finite fields)
create Mestre’s conic from the Igusa-Clebsch invariants
AUTHORS:
Florian Bouyer
Marco Streng
- sage.schemes.hyperelliptic_curves.mestre.HyperellipticCurve_from_invariants(i, reduced=True, precision=None, algorithm='default')#
Returns a hyperelliptic curve with the given Igusa-Clebsch invariants up to scaling.
The output is a curve over the field in which the Igusa-Clebsch invariants are given. The output curve is unique up to isomorphism over the algebraic closure. If no such curve exists over the given field, then raise a ValueError.
INPUT:
i
- list or tuple of length 4 containing the four Igusa-Clebsch invariants: I2,I4,I6,I10.reduced
- Boolean (default = True) If True, tries to reduce the polynomial defining the hyperelliptic curve using the functionreduce_polynomial()
(see thereduce_polynomial()
documentation for more details).precision
- integer (default = None) Which precision for real and complex numbers should the reduction use. This only affects the reduction, not the correctness. If None, the algorithm uses the default 53 bit precision.algorithm
-'default'
or'magma'
. If set to'magma'
, uses Magma to parameterize Mestre’s conic (needs Magma to be installed).
OUTPUT:
A hyperelliptic curve object.
EXAMPLES:
Examples over the rationals:
sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856]) Traceback (most recent call last): ... NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756. sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856],reduced = False) Hyperelliptic Curve over Rational Field defined by y^2 = -46656*x^6 + 46656*x^5 - 19440*x^4 + 4320*x^3 - 540*x^2 + 4410*x - 1 sage: HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1]) Traceback (most recent call last): ... NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756.
An example over a finite field:
sage: H = HyperellipticCurve_from_invariants([GF(13)(1),3,7,5]); H Hyperelliptic Curve over Finite Field of size 13 defined by ... sage: H.igusa_clebsch_invariants() (4, 9, 6, 11)
An example over a number field:
sage: K = QuadraticField(353, 'a') sage: H = HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1], reduced = false) sage: f = K['x'](H.hyperelliptic_polynomials()[0])
If the Mestre Conic defined by the Igusa-Clebsch invariants has no rational points, then there exists no hyperelliptic curve over the base field with the given invariants.:
sage: HyperellipticCurve_from_invariants([1,2,3,4]) Traceback (most recent call last): ... ValueError: No such curve exists over Rational Field as there are no rational points on Projective Conic Curve over Rational Field defined by -2572155000*u^2 - 317736000*u*v + 1250755459200*v^2 + 2501510918400*u*w + 39276887040*v*w + 2736219686912*w^2
Mestre’s algorithm only works for generic curves of genus two, so another algorithm is needed for those curves with extra automorphism. See also trac ticket #12199:
sage: P.<x> = QQ[] sage: C = HyperellipticCurve(x^6+1) sage: i = C.igusa_clebsch_invariants() sage: HyperellipticCurve_from_invariants(i) Traceback (most recent call last): ... TypeError: F (=0) must have degree 2
Igusa-Clebsch invariants also only work over fields of characteristic different from 2, 3, and 5, so another algorithm will be needed for fields of those characteristics. See also trac ticket #12200:
sage: P.<x> = GF(3)[] sage: HyperellipticCurve(x^6+x+1).igusa_clebsch_invariants() Traceback (most recent call last): ... NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5 sage: HyperellipticCurve_from_invariants([GF(5)(1),1,0,1]) Traceback (most recent call last): ... ZeroDivisionError: inverse of Mod(0, 5) does not exist
ALGORITHM:
This is Mestre’s algorithm [Mes1991]. Our implementation is based on the formulae on page 957 of [LY2001], cross-referenced with [Wam1999b] to correct typos.
First construct Mestre’s conic using the
Mestre_conic()
function. Parametrize the conic if possible. Let \(f_1, f_2, f_3\) be the three coordinates of the parametrization of the conic by the projective line, and change them into one variable by letting \(F_i = f_i(t, 1)\). Note that each \(F_i\) has degree at most 2.Then construct a sextic polynomial \(f = \sum_{0<=i,j,k<=3}{c_{ijk}*F_i*F_j*F_k}\), where \(c_{ijk}\) are defined as rational functions in the invariants (see the source code for detailed formulae for \(c_{ijk}\)). The output is the hyperelliptic curve \(y^2 = f\).
- sage.schemes.hyperelliptic_curves.mestre.Mestre_conic(i, xyz=False, names='u,v,w')#
Return the conic equation from Mestre’s algorithm given the Igusa-Clebsch invariants.
It has a rational point if and only if a hyperelliptic curve corresponding to the invariants exists.
INPUT:
i
- list or tuple of length 4 containing the four Igusa-Clebsch invariants: I2, I4, I6, I10xyz
- Boolean (default: False) if True, the algorithm also returns three invariants x,y,z used in Mestre’s algorithmnames
(default: ‘u,v,w’) - the variable names for the Conic
OUTPUT:
A Conic object
EXAMPLES:
A standard example:
sage: Mestre_conic([1,2,3,4]) Projective Conic Curve over Rational Field defined by -2572155000*u^2 - 317736000*u*v + 1250755459200*v^2 + 2501510918400*u*w + 39276887040*v*w + 2736219686912*w^2
Note that the algorithm works over number fields as well:
sage: k = NumberField(x^2-41,'a') sage: a = k.an_element() sage: Mestre_conic([1,2+a,a,4+a]) Projective Conic Curve over Number Field in a with defining polynomial x^2 - 41 defined by (-801900000*a + 343845000)*u^2 + (855360000*a + 15795864000)*u*v + (312292800000*a + 1284808579200)*v^2 + (624585600000*a + 2569617158400)*u*w + (15799910400*a + 234573143040)*v*w + (2034199306240*a + 16429854656512)*w^2
And over finite fields:
sage: Mestre_conic([GF(7)(10),GF(7)(1),GF(7)(2),GF(7)(3)]) Projective Conic Curve over Finite Field of size 7 defined by -2*u*v - v^2 - 2*u*w + 2*v*w - 3*w^2
An example with xyz:
sage: Mestre_conic([5,6,7,8], xyz=True) (Projective Conic Curve over Rational Field defined by -415125000*u^2 + 608040000*u*v + 33065136000*v^2 + 66130272000*u*w + 240829440*v*w + 10208835584*w^2, 232/1125, -1072/16875, 14695616/2109375)
ALGORITHM:
The formulas are taken from pages 956 - 957 of [LY2001] and based on pages 321 and 332 of [Mes1991].
See the code or [LY2001] for the detailed formulae defining x, y, z and L.