Module: sage.interfaces.genus2reduction
Conductor and Reduction Types for Genus 2 Curves
Author Log:
ACKNOWLEDGMENT: (From Liu's website:) Many thanks to Henri Cohen who started writing this program. After this program is available, many people pointed out to me (mathematical as well as programming) bugs : B. Poonen, E. Schaefer, C. Stahlke, M. Stoll, F. Villegas. So thanks to all of them. Thanks also go to Ph. Depouilly who help me to compile the program.
Also Liu has given me explicit permission to include genus2reduction with SAGE and for people to modify the C source code however they want.
Module-level Functions
) |
Class: Genus2reduction
Use R = genus2reduction(Q, P)
to obtain reduction
information about the Jacobian of the projective smooth curve
defined by
. Type
R?
for further
documentation and a description of how to interpret the local
reduction data.
sage: x = QQ['x'].0 sage: R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5) sage: R.conductor 1416875 sage: factor(R.conductor) 5^4 * 2267
This means that only the odd part of the conductor is known.
sage: R.prime_to_2_conductor_only True
The discriminant is always minimal away from 2, but possibly not at 2.
sage: factor(R.minimal_disc) 2^3 * 5^5 * 2267
Printing R summarizes all the information computed about the curve
sage: R Reduction data about this proper smooth genus 2 curve: y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5 A Minimal Equation (away from 2): y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855 Minimal Discriminant (away from 2): 56675000 Conductor (away from 2): 1416875 Local Data: p=2 (potential) stable reduction: (II), j=1 p=5 (potential) stable reduction: (I) reduction at p: [V] page 156, (3), f=4 p=2267 (potential) stable reduction: (II), j=432 reduction at p: [I{1-0-0}] page 170, (1), f=1
Here are some examples of curves with modular Jacobians:
sage: R = genus2reduction(x^3 + x + 1, -2*x^5 - 3*x^2 + 2*x - 2) sage: factor(R.conductor) 23^2 sage: factor(genus2reduction(x^3 + 1, -x^5 - 3*x^4 + 2*x^2 + 2*x - 2).conductor) 29^2 sage: factor(genus2reduction(x^3 + x + 1, x^5 + 2*x^4 + 2*x^3 + x^2 - x - 1).conductor) 5^6
sage: genus2reduction(0, x^6 + 3*x^3 + 63) Reduction data about this proper smooth genus 2 curve: y^2 = x^6 + 3*x^3 + 63 A Minimal Equation (away from 2): y^2 = x^6 + 3*x^3 + 63 Minimal Discriminant (away from 2): 10628388316852992 Conductor (away from 2): 2893401 Local Data: p=2 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 p=3 (potential) stable reduction: (I) reduction at p: [III{9}] page 184, (3)^2, f=10 p=7 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 reduction at p: [I{0}-II-0] page 159, (1), f=2
In the above example, Liu remarks that in fact at
, the reduction
is [II-II-0] page 163, (1),
. So the conductor of J(C) is actually
.
A MODULAR CURVE:
Consider the modular curve
defined by an equation
We have:
sage: genus2reduction(x^3-x^2-1, x^2 - x) Reduction data about this proper smooth genus 2 curve: y^2 + (x^3 - x^2 - 1)*y = x^2 - x A Minimal Equation (away from 2): y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561 Minimal Discriminant (away from 2): 169 Conductor: 169 Local Data: p=13 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 reduction at p: [I{0}-II-0] page 159, (1), f=2
So the curve has good reduction at 2. At
, the stable reduction is
union of two elliptic curves, and both of them have 0 as modular invariant.
The reduction at 13 is of type [I_0-II-0] (see Namikawa-Ueno,
page 159). It is an elliptic curve with a cusp. The group of connected
components of the Neron model of
is trivial, and the exponent
of the conductor of
at
is
. The conductor of
is
. (Note: It is a theorem of Conrad-Edixhoven-Stein that the
component group of
is trivial for all primes
.)
self) |
Functions: console,
raw
self, Q, P) |
Return the raw output of running the genus2reduction
program on the hyperelliptic curve
as a
string.
Input:
sage: x = QQ['x'].0 sage: print genus2reduction.raw(x^3 - 2*x^2 - 2*x + 1, -5*x^5)[0] a minimal equation over Z[1/2] is : y^2 = x^6-240*x^4-2550*x^3-11400*x^2-24100*x-19855 factorization of the minimal (away from 2) discriminant : [2,3;5,5;2267,1] p=2 (potential) stable reduction : (II), j=1 p=5 (potential) stable reduction : (I) reduction at p : [V] page 156, (3), f=4 p=2267 (potential) stable reduction : (II), j=432 reduction at p : [I{1-0-0}] page 170, (1), f=1 the prime to 2 part of the conductor is 1416875 in factorized form : [2,0;5,4;2267,1]
Special Functions: __call__,
__init__,
__reduce__,
_repr_
Class: Genus2reduction_expect
self, [server=None], [server_tmpdir=None], [logfile=None]) |
Special Functions: __init__
Class: ReductionData
How to read local_data
attribute, i.e., if this class is R,
then the following is the meaning of R.local_data[p]
.
For each prime number
dividing the discriminant of
,
there are two lines.
The first line contains information about the stable reduction after field extension. Here are the meanings of the symbols of stable reduction :
(I) The stable reduction is smooth (i.e. the curve has potentially good reduction).
(II) The stable reduction is an elliptic curve
with an ordinary double
point.
mod
is the modular invariant of
.
(III) The stable reduction is a projective line with two ordinary double points.
(IV) The stable reduction is two projective lines crossing transversally at three points.
(V) The stable reduction is the union of two elliptic curves
and
intersecting transversally at one point. Let
,
be their modular
invariants, then
and
are computed (they are numbers
mod
).
(VI) The stable reduction is the union of an elliptic curve
and a
projective line which has an ordinary double point. These two
components intersect transversally at one point.
mod
is the
modular invariant of
.
(VII) The stable reduction is as above, but the two components are both singular.
In the cases (I) and (V), the Jacobian
has potentially good
reduction. In the cases (III), (IV) and (VII),
has
potentially multiplicative reduction. In the two remaining cases,
the (potential) semi-abelian reduction of
is extension of an
elliptic curve (with modular invariant
mod
) by a torus.
The second line contains three data concerning the reduction at
without any field extension.
Hn
is
isomorphic to (2)x(2) if n is even and to (4) otherwise.
Note - The set of rational points of
can be
computed using Theorem 1.17 in S. Bosch and Q. Liu
"Rational points of the group of components of a
Néron model", Manuscripta Math. 98 (1999), 275-293.
Note:
Warning - Be careful regarding the formula:
. The fact is that the minimal
discriminant may change after unramified extension.
One can show however that, at worst, the change will
stabilize after a quadratic unramified extension
(Q. Liu : "Modeles entiers de courbes hyperelliptiques sur un
corps de valuation discrete", Trans. AMS 348 (1996), 4577-4610,
§7.2, Proposition 4).
self, raw, P, Q, minimal_equation, minimal_disc, local_data, conductor, prime_to_2_conductor_only) |
Special Functions: __init__,
_local_data_str,
_repr_
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