Conductor and Reduction Types for Genus 2 Curves

AUTHORS:

  • Qing Liu and Henri Cohen (1994-1998): wrote genus2reduction C program
  • William Stein (2006-03-05): wrote Sage interface to genus2reduction
  • Jeroen Demeyer (2014-09-17): replace genus2reduction program by PARI library call (trac ticket #15808)

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.

class sage.interfaces.genus2reduction.Genus2reduction

Bases: sage.structure.sage_object.SageObject

Conductor and Reduction Types for Genus 2 Curves.

Use R = genus2reduction(Q, P) to obtain reduction information about the Jacobian of the projective smooth curve defined by \(y^2 + Q(x)y = P(x)\). Type R? for further documentation and a description of how to interpret the local reduction data.

EXAMPLES:

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

EXAMPLE:

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 \(p=2\), the reduction is [II-II-0] page 163, (1), \(f=8\). So the conductor of J(C) is actually \(2 \cdot 2893401=5786802\).

A MODULAR CURVE:

Consider the modular curve \(X_1(13)\) defined by an equation

\[y^2 + (x^3-x^2-1)y = x^2 - x.\]

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 \(p=13\), 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 \(J(C)\) is trivial, and the exponent of the conductor of \(J(C)\) at \(13\) is \(f=2\). The conductor of \(J(C)\) is \(13^2\). (Note: It is a theorem of Conrad-Edixhoven-Stein that the component group of \(J(X_1(p))\) is trivial for all primes \(p\).)

raw(Q, P)

Return a string emulating the raw output of running the old genus2reduction program on the hyperelliptic curve \(y^2 + Q(x)y = P(x)\).

INPUT:

  • Q - something coercible to a univariate polynomial over Q.
  • P - something coercible to a univariate polynomial over Q.

OUTPUT:

  • string - raw output
  • Q - what Q was actually input to auxiliary genus2reduction program
  • P - what P was actually input to auxiliary genus2reduction program

EXAMPLES:

sage: x = QQ['x'].0
sage: print genus2reduction.raw(x^3 - 2*x^2 - 2*x + 1, -5*x^5)[0]
doctest:...: DeprecationWarning: the raw() method is provided for backwards compatibility only, use the result of the genus2reduction() call instead of parsing strings
See http://trac.sagemath.org/15808 for details.
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 : [5,4;2267,1]
class sage.interfaces.genus2reduction.ReductionData(pari_result, P, Q, minimal_equation, minimal_disc, local_data, conductor, prime_to_2_conductor_only)

Bases: sage.structure.sage_object.SageObject

Reduction data for a genus 2 curve.

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 \(p\) dividing the discriminant of \(y^2+Q(x)y=P(x)\), 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 \(E\) with an ordinary double point. \(j\) mod \(p\) is the modular invariant of \(E\).

(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 \(E_1\) and \(E_2\) intersecting transversally at one point. Let \(j_1\), \(j_2\) be their modular invariants, then \(j_1+j_2\) and \(j_1 j_2\) are computed (they are numbers mod \(p\)).

(VI) The stable reduction is the union of an elliptic curve \(E\) and a projective line which has an ordinary double point. These two components intersect transversally at one point. \(j\) mod \(p\) is the modular invariant of \(E\).

(VII) The stable reduction is as above, but the two components are both singular.

In the cases (I) and (V), the Jacobian \(J(C)\) has potentially good reduction. In the cases (III), (IV) and (VII), \(J(C)\) has potentially multiplicative reduction. In the two remaining cases, the (potential) semi-abelian reduction of \(J(C)\) is extension of an elliptic curve (with modular invariant \(j\) mod \(p\)) by a torus.

The second line contains three data concerning the reduction at \(p\) without any field extension.

  1. The first symbol describes the REDUCTION AT \(p\) of \(C\). We use the symbols of Namikawa-Ueno for the type of the reduction (Namikawa, Ueno:”The complete classification of fibers in pencils of curves of genus two”, Manuscripta Math., vol. 9, (1973), pages 143-186.) The reduction symbol is followed by the corresponding page number (or just an indiction) in the above article. The lower index is printed by , for instance, [I2-II-5] means [I_2-II-5]. Note that if \(K\) and \(K'\) are Kodaira symbols for singular fibers of elliptic curves, [K-K’-m] and [K’-K-m] are the same type. Finally, [K-K’-1] (not the same as [K-K’-1]) is [K’-K-alpha] in the notation of Namikawa-Ueno. The figure [2I_0-m] in Namikawa-Ueno, page 159 must be denoted by [2I_0-(m+1)].

  2. The second datum is the GROUP OF CONNECTED COMPONENTS (over an ALGEBRAIC CLOSURE (!) of \(\GF{p}\)) of the Neron model of J(C). The symbol (n) means the cyclic group with n elements. When n=0, (0) is the trivial group (1). Hn is isomorphic to (2)x(2) if n is even and to (4) otherwise.

    Note - The set of rational points of \(\Phi\) can be computed using Theorem 1.17 in S. Bosch and Q. Liu “Rational points of the group of components of a Neron model”, Manuscripta Math. 98 (1999), 275-293.

  3. Finally, \(f\) is the exponent of the conductor of \(J(C)\) at \(p\).

Warning

Be careful regarding the formula:

\[\text{valuation of the naive minimal discriminant} = f + n - 1 + 11c(X).\]

(Q. Liu : “Conducteur et discriminant minimal de courbes de genre 2”, Compositio Math. 94 (1994) 51-79, Theoreme 2) is valid only if the residual field is algebraically closed as stated in the paper. So this equality does not hold in general over \(\QQ_p\). 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, Section 7.2, Proposition 4).

sage.interfaces.genus2reduction.divisors_to_string(divs)

Convert a list of numbers (representing the orders of cyclic groups in the factorization of a finite abelian group) to a string according to the format shown in the examples.

INPUT:

  • divs – a (possibly empty) list of numbers

OUTPUT: a string representation of these numbers

EXAMPLES:

sage: from sage.interfaces.genus2reduction import divisors_to_string
sage: print divisors_to_string([])
(1)
sage: print divisors_to_string([5])
(5)
sage: print divisors_to_string([5]*6)
(5)^6
sage: print divisors_to_string([2,3,4])
(2)x(3)x(4)
sage: print divisors_to_string([6,2,2])
(6)x(2)^2

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