************* genus2reduction *************
November 7th, 1994
WHAT THIS PROGRAM DOES.............................................
Let C be a proper smooth curve of genus 2 defined by a hyperelliptic
equation
y^2+Q(x)y=P(x)
where P(x) and Q(x) are polynomials with rational coefficients such
that deg(Q(x))<4, deg(P(x))<7.
Let J(C) be the Jacobian of C, let X be the minimal regular model of
C over the ring of integers Z.
This program determines the reduction of C at any prime number p (that
is the special fiber X_p of X over p), and the exponent f of the conductor
of J(C) at p.
Unfortunately, this program is not yet complete for p=2.
HOW TO RUN THIS PROGRAM............................................
After you compile successfully genus2reduction,
type genus2reduction and enter. You will be asked to enter the
polynomials Q(x) and P(x) (Example: x^3-2*x^2-2*x+1 for Q(x)
and -5*x^5 for P(x). Don't leave space in between two terms in a
polynomial).
You then get a minimal equation over Z[1/2], the factorization
of (the absolute value of) its discriminant (called naive minimal
discriminant). For each prime number p dividing the discriminant
of the initial equation y^2+Q(x)*y=P(x), some data concerning the
reduction mod p are listed (see below). Finally the prime to 2 part
of the conductor of J(C) is given. It is just the product of the local
terms p^f. In some cases, the conductor itself is computed.
Entering 0 for both Q(x) and P(x) will exit normally the program.
You can type Ctrl C to interrupt the program.
HOW TO READ THE RESULTS.................................................
For each prime number p dividing the discriminant of y^2+Q(x)*y=P(x), one
gets the results in 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 j1, j2 be their modular
invariants, then j1+j2 and j1*j2 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.
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, [I{2}-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)].
The second datum is the group of connected components (over an algebraic
closure of F_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). H{n} is isomorphic to (2)x(2) if n is even and to
(4) otherwise.
Finally, f is the exponent of the conductor of J(C) at p.
TEST EXAMPLES................................................
1.
Consider the curve defined by y^2=x^6+3*x^3+63.
Run genus2reduction and enter 0 for Q(x), x^6+3*x^3+63 for P(x).
Then you get :
a minimal equation over Z[1/2] is :
y^2 = x^6 + 3*x^3 + 63
factorization of the minimal (away from 2) discriminant :
[2, 8; 3, 25; 7, 2]
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
the prime to 2 part of the conductor is 2893401
It can be seen that at p=2, the reduction is [II-II-0] page 163, (1), f=8.
So the conductor of J(C) is 2*2893401=5786802.
2.
Consider the modular curve X_1(13) defined by an equation
y^2+(x^3-x^2-1)*y=x^2-x
Run genus2reduction, and enter x^3-x^2-1 for Q(x) and x^2-x for
P(x). Then you get
a minimal equation over Z[1/2] is :
y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
factorization of the minimal (away from 2) discriminant :
[13, 2]
p=13
(potential) stable reduction : (V), j1+j2=0, j1*j2=0
reduction at p : [I{0}-II-0}] page 159, (1), f=2
the conductor is 169
So the curve has good reduction at 2. At p=13, the stable reduction is
union of two elliptic curves, both of them have 0 as modular invariant.
The reduction at 13 is of type [I_0-II-0] (see Namikawa-Ueno, op. cit,
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.
REMARKS..................................................................
This program is based entirely on Pari (developed by C. Batut, D. Bernardi,
H. Cohen and M. Olivier). For small primes 3, 5, 7, it has been tested at
least twice for each type of reduction listed in Namikawa-Ueno (op. cit.).
But it doesn't exclude bugs. Please report any problem or bug you could find
to :
liu@math.u-bordeaux.fr
If you get this program by ftp, please send a message to the above
address. You will be informed if there are further developments (especially
concerning the reduction at p=2).
Qing LIU
CNRS, Laboratoire de Mathematiques Pures
Universite de Bordeaux 1
351, cours de la Liberation
33405 Talence cedex
FRANCE