************* genus2reduction *************12November 7th, 19943456WHAT THIS PROGRAM DOES.............................................78Let C be a proper smooth curve of genus 2 defined by a hyperelliptic9equation1011y^2+Q(x)y=P(x)1213where P(x) and Q(x) are polynomials with rational coefficients such14that deg(Q(x))<4, deg(P(x))<7.1516Let J(C) be the Jacobian of C, let X be the minimal regular model of17C over the ring of integers Z.1819This program determines the reduction of C at any prime number p (that20is the special fiber X_p of X over p), and the exponent f of the conductor21of J(C) at p.2223Unfortunately, this program is not yet complete for p=2.24252627HOW TO RUN THIS PROGRAM............................................2829After you compile successfully genus2reduction,30type genus2reduction and enter. You will be asked to enter the31polynomials Q(x) and P(x) (Example: x^3-2*x^2-2*x+1 for Q(x)32and -5*x^5 for P(x). Don't leave space in between two terms in a33polynomial).3435You then get a minimal equation over Z[1/2], the factorization36of (the absolute value of) its discriminant (called naive minimal37discriminant). For each prime number p dividing the discriminant38of the initial equation y^2+Q(x)*y=P(x), some data concerning the39reduction mod p are listed (see below). Finally the prime to 2 part40of the conductor of J(C) is given. It is just the product of the local41terms p^f. In some cases, the conductor itself is computed.4243Entering 0 for both Q(x) and P(x) will exit normally the program.4445You can type Ctrl C to interrupt the program.46474849HOW TO READ THE RESULTS.................................................5051For each prime number p dividing the discriminant of y^2+Q(x)*y=P(x), one52gets the results in two lines.5354The first line contains information about the stable reduction after55field extension. Here are the meanings of the symbols of stable reduction :5657(I) The stable reduction is smooth (i.e. the curve has potentially58good reduction).59(II) The stable reduction is an elliptic curve E with an ordinary double60point. j mod p is the modular invariant of E.61(III) The stable reduction is a projective line with two ordinary double62points.63(IV) The stable reduction is two projective lines crossing transversally64at three points.65(V) The stable reduction is the union of two elliptic curves E_1 and E_266intersecting transversally at one point. Let j1, j2 be their modular67invariants, then j1+j2 and j1*j2 are computed (they are numbers mod p).68(VI) The stable reduction is the union of an elliptic curve E and a69projective line which has an ordinary double point. These two70components intersect transversally at one point. j mod p is the71modular invariant of E.72(VII) The stable reduction is as above, but the two components are both73singular.7475In the cases (I) and (V), the Jacobian J(C) has potentially good reduction.76In the cases (III), (IV) and (VII), J(C) has potentially multiplicative77reduction. In the two remaining cases, the (potential) semi-abelian78reduction of J(C) is extension of an elliptic curve (with modular invariant79j mod p) by a torus.8081The second line contains three data concerning the reduction at p without82any field extension.83The first symbol describes the reduction at p of C. We use the symbols of84Namikawa-Ueno for the type of the reduction (Namikawa, Ueno : "The complete85classification of fibers in pencils of curves of genus two", Manuscripta86Math., vol. 9, (1973), pages 143-186.) The reduction symbol is followed by87the corresponding page number (or just an indiction) in the above article.88The lower index is printed by { }, for instance, [I{2}-II-5] means [I_2-II-5].89Note that if K and K' are Kodaira symbols for singular fibers of elliptic90curves, [K-K'-m] and [K'-K-m] are the same type. Finally, [K-K'--1] (not the91same as [K-K'-1]) is [K'-K-alpha] in the notation of Namikawa-Ueno. The figure92[2I_0-m] in Namikawa-Ueno, page 159 must be denoted by [2I_0-(m+1)].93The second datum is the group of connected components (over an algebraic94closure of F_p) of the Neron model of J(C). The symbol (n) means the cyclic95group with n elements. When n=0, (0) is96the trivial group (1). H{n} is isomorphic to (2)x(2) if n is even and to97(4) otherwise.98Finally, f is the exponent of the conductor of J(C) at p.99100101TEST EXAMPLES................................................1021031.104Consider the curve defined by y^2=x^6+3*x^3+63.105Run genus2reduction and enter 0 for Q(x), x^6+3*x^3+63 for P(x).106Then you get :107108a minimal equation over Z[1/2] is :109y^2 = x^6 + 3*x^3 + 63110111factorization of the minimal (away from 2) discriminant :112[2, 8; 3, 25; 7, 2]113114p=2115(potential) stable reduction : (V), j1+j2=0, j1*j2=0116p=3117(potential) stable reduction : (I)118reduction at p : [III{9}] page 184, (3)^2, f=10119p=7120(potential) stable reduction : (V), j1+j2=0, j1*j2=0121reduction at p : [I{0}-II-0}] page 159, (1), f=2122123the prime to 2 part of the conductor is 2893401124125It can be seen that at p=2, the reduction is [II-II-0] page 163, (1), f=8.126So the conductor of J(C) is 2*2893401=5786802.1271282.129Consider the modular curve X_1(13) defined by an equation130y^2+(x^3-x^2-1)*y=x^2-x131Run genus2reduction, and enter x^3-x^2-1 for Q(x) and x^2-x for132P(x). Then you get133134a minimal equation over Z[1/2] is :135y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561136137factorization of the minimal (away from 2) discriminant :138[13, 2]139140p=13141(potential) stable reduction : (V), j1+j2=0, j1*j2=0142reduction at p : [I{0}-II-0}] page 159, (1), f=2143144the conductor is 169145146So the curve has good reduction at 2. At p=13, the stable reduction is147union of two elliptic curves, both of them have 0 as modular invariant.148The reduction at 13 is of type [I_0-II-0] (see Namikawa-Ueno, op. cit,149page 159). It is an elliptic curve with a cusp. The group of connected150components of the Neron model of J(C) is trivial, and the exponent151of the conductor of J(C) at 13 is f=2. The conductor of J(C) is 13^2.152153154REMARKS..................................................................155156This program is based entirely on Pari (developed by C. Batut, D. Bernardi,157H. Cohen and M. Olivier). For small primes 3, 5, 7, it has been tested at158least twice for each type of reduction listed in Namikawa-Ueno (op. cit.).159But it doesn't exclude bugs. Please report any problem or bug you could find160to :161162[email protected]163164If you get this program by ftp, please send a message to the above165address. You will be informed if there are further developments (especially166concerning the reduction at p=2).167168Qing LIU169CNRS, Laboratoire de Mathematiques Pures170Universite de Bordeaux 1171351, cours de la Liberation17233405 Talence cedex173FRANCE174175176