Sharedwww / tables / genus2reduction / README.txtOpen in CoCalc
Author: William A. Stein
1*************            genus2reduction             *************
2
3                         November 7th, 1994
4
5
6
7WHAT THIS PROGRAM DOES.............................................
8
9Let C be a proper smooth curve of genus 2 defined by a hyperelliptic
10equation
11
12                         y^2+Q(x)y=P(x)
13
14where P(x) and Q(x) are polynomials with rational coefficients such
15that deg(Q(x))<4, deg(P(x))<7.
16
17Let J(C) be the Jacobian of C, let X be the minimal regular model of
18C over the ring of integers Z.
19
20This program determines the reduction of C at any prime number p (that
21is the special fiber X_p of X over p), and the exponent f of the conductor
22of J(C) at p.
23
24Unfortunately, this program is not yet complete for p=2.
25
26
27
28HOW TO RUN THIS PROGRAM............................................
29
30After you compile successfully genus2reduction,
31type genus2reduction and enter. You will be asked to enter the
32polynomials Q(x) and P(x) (Example: x^3-2*x^2-2*x+1 for Q(x)
33and -5*x^5 for P(x). Don't leave space in between two terms in a
34polynomial).
35
36You then get a minimal equation over Z[1/2], the factorization
37of (the absolute value of) its discriminant (called naive minimal
38discriminant). For each prime number p dividing the discriminant
39of the initial equation y^2+Q(x)*y=P(x), some data concerning the
40reduction mod p are listed (see below). Finally the prime to 2 part
41of the conductor of J(C) is given. It is just the product of the local
42terms p^f. In some cases, the conductor itself is computed.
43
44Entering 0 for both Q(x) and P(x) will exit normally the program.
45
46You can type Ctrl C to interrupt the program.
47
48
49
51
52For each prime number p dividing the discriminant of y^2+Q(x)*y=P(x), one
53gets the results in two lines.
54
55The first line contains information about the stable reduction after
56field extension. Here are the meanings of the symbols of stable reduction :
57
58(I)   The stable reduction is smooth (i.e. the curve has potentially
59      good reduction).
60(II)  The stable reduction is an elliptic curve E with an ordinary double
61      point. j mod p is the modular invariant of E.
62(III) The stable reduction is a projective line with two ordinary double
63      points.
64(IV)  The stable reduction is two projective lines crossing transversally
65      at three points.
66(V)   The stable reduction is the union of two elliptic curves E_1 and E_2
67      intersecting transversally at one point. Let j1, j2 be their modular
68      invariants, then j1+j2 and j1*j2 are computed (they are numbers mod p).
69(VI)  The stable reduction is the union of an elliptic curve E and a
70      projective line which has an ordinary double point. These two
71      components intersect transversally at one point. j mod p is the
72      modular invariant of E.
73(VII) The stable reduction is as above, but the two components are both
74      singular.
75
76In the cases (I) and (V), the Jacobian J(C) has potentially good reduction.
77In the cases (III), (IV) and (VII), J(C) has potentially multiplicative
78reduction. In the two remaining cases, the (potential) semi-abelian
79reduction of J(C) is extension of an elliptic curve (with modular invariant
80j mod p) by a torus.
81
82The second line contains three data concerning the reduction at p without
83any field extension.
84The first symbol describes the reduction at p of C. We use the symbols of
85Namikawa-Ueno for the type of the reduction (Namikawa, Ueno : "The complete
86classification of fibers in pencils of curves of genus two", Manuscripta
87Math., vol. 9, (1973), pages 143-186.) The reduction symbol is followed by
88the corresponding page number (or just an indiction) in the above article.
89The lower index is printed by { }, for instance, [I{2}-II-5] means [I_2-II-5].
90Note that if K and K' are Kodaira symbols for singular fibers of elliptic
91curves, [K-K'-m] and [K'-K-m] are the same type. Finally, [K-K'--1] (not the
92same as [K-K'-1]) is [K'-K-alpha] in the notation of Namikawa-Ueno. The figure
93[2I_0-m] in Namikawa-Ueno, page 159 must be denoted by [2I_0-(m+1)].
94The second datum is the group of connected components (over an algebraic
95closure of F_p) of the Neron model of J(C). The symbol (n) means the cyclic
96group with n elements. When n=0, (0) is
97the trivial group (1). H{n} is isomorphic to (2)x(2) if n is even and to
98(4) otherwise.
99Finally, f is the exponent of the conductor of J(C) at p.
100
101
102TEST EXAMPLES................................................
103
1041.
105Consider the curve defined by y^2=x^6+3*x^3+63.
106Run genus2reduction and enter 0 for Q(x), x^6+3*x^3+63 for P(x).
107Then you get :
108
109     a minimal equation over Z[1/2] is :
110     y^2 = x^6 + 3*x^3 + 63
111
112     factorization of the minimal (away from 2) discriminant :
113     [2, 8; 3, 25; 7, 2]
114
115     p=2
116     (potential) stable reduction :  (V), j1+j2=0, j1*j2=0
117     p=3
118     (potential) stable reduction :  (I)
119     reduction at p : [III{9}] page 184, (3)^2, f=10
120     p=7
121     (potential) stable reduction :  (V), j1+j2=0, j1*j2=0
122     reduction at p : [I{0}-II-0}] page 159, (1), f=2
123
124     the prime to 2 part of the conductor is 2893401
125
126It can be seen that at p=2, the reduction is [II-II-0] page 163, (1), f=8.
127So the conductor of J(C) is 2*2893401=5786802.
128
1292.
130Consider the modular curve X_1(13) defined by an equation
131            y^2+(x^3-x^2-1)*y=x^2-x
132Run genus2reduction, and enter x^3-x^2-1 for Q(x) and x^2-x for
133P(x). Then you get
134
135     a minimal equation over Z[1/2] is :
136     y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
137
138     factorization of the minimal (away from 2) discriminant :
139     [13, 2]
140
141     p=13
142     (potential) stable reduction :  (V), j1+j2=0, j1*j2=0
143     reduction at p : [I{0}-II-0}] page 159, (1), f=2
144
145     the conductor is 169
146
147So the curve has good reduction at 2. At p=13, the stable reduction is
148union of two elliptic curves, both of them have 0 as modular invariant.
149The reduction at 13 is of type [I_0-II-0] (see Namikawa-Ueno, op. cit,
150page 159). It is an elliptic curve with a cusp. The group of connected
151components of the Neron model of J(C) is trivial, and the exponent
152of the conductor of J(C) at 13 is f=2. The conductor of J(C) is 13^2.
153
154
155REMARKS..................................................................
156
157This program is based entirely on Pari (developed by C. Batut, D. Bernardi,
158H. Cohen and M. Olivier). For small primes 3, 5, 7, it has been tested at
159least twice for each type of reduction listed in Namikawa-Ueno (op. cit.).
160But it doesn't exclude bugs. Please report any problem or bug you could find
161to :
162
163[email protected]
164
165If you get this program by ftp, please send a message to the above
166address. You will be informed if there are further developments (especially
167concerning the reduction at p=2).
168
169Qing LIU
170CNRS, Laboratoire de Mathematiques Pures
171Universite de Bordeaux 1
172351, cours de la Liberation
17333405 Talence cedex
174FRANCE
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