************* 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