Sharedwww / Tables / genus2reduction / README.txtOpen in CoCalc
Author: William A. Stein
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************* genus2reduction *************
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November 7th, 1994
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WHAT THIS PROGRAM DOES.............................................
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Let C be a proper smooth curve of genus 2 defined by a hyperelliptic
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equation
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y^2+Q(x)y=P(x)
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where P(x) and Q(x) are polynomials with rational coefficients such
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that deg(Q(x))<4, deg(P(x))<7.
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Let J(C) be the Jacobian of C, let X be the minimal regular model of
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C over the ring of integers Z.
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This program determines the reduction of C at any prime number p (that
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is the special fiber X_p of X over p), and the exponent f of the conductor
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of J(C) at p.
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Unfortunately, this program is not yet complete for p=2.
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HOW TO RUN THIS PROGRAM............................................
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After you compile successfully genus2reduction,
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type genus2reduction and enter. You will be asked to enter the
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polynomials Q(x) and P(x) (Example: x^3-2*x^2-2*x+1 for Q(x)
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and -5*x^5 for P(x). Don't leave space in between two terms in a
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polynomial).
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You then get a minimal equation over Z[1/2], the factorization
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of (the absolute value of) its discriminant (called naive minimal
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discriminant). For each prime number p dividing the discriminant
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of the initial equation y^2+Q(x)*y=P(x), some data concerning the
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reduction mod p are listed (see below). Finally the prime to 2 part
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of the conductor of J(C) is given. It is just the product of the local
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terms p^f. In some cases, the conductor itself is computed.
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Entering 0 for both Q(x) and P(x) will exit normally the program.
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You can type Ctrl C to interrupt the program.
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HOW TO READ THE RESULTS.................................................
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For each prime number p dividing the discriminant of y^2+Q(x)*y=P(x), one
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gets the results in two lines.
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The first line contains information about the stable reduction after
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field extension. Here are the meanings of the symbols of stable reduction :
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(I) The stable reduction is smooth (i.e. the curve has potentially
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good reduction).
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(II) The stable reduction is an elliptic curve E with an ordinary double
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point. j mod p is the modular invariant of E.
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(III) The stable reduction is a projective line with two ordinary double
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points.
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(IV) The stable reduction is two projective lines crossing transversally
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at three points.
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(V) The stable reduction is the union of two elliptic curves E_1 and E_2
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intersecting transversally at one point. Let j1, j2 be their modular
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invariants, then j1+j2 and j1*j2 are computed (they are numbers mod p).
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(VI) The stable reduction is the union of an elliptic curve E and a
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projective line which has an ordinary double point. These two
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components intersect transversally at one point. j mod p is the
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modular invariant of E.
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(VII) The stable reduction is as above, but the two components are both
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singular.
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In the cases (I) and (V), the Jacobian J(C) has potentially good reduction.
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In the cases (III), (IV) and (VII), J(C) has potentially multiplicative
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reduction. In the two remaining cases, the (potential) semi-abelian
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reduction of J(C) is extension of an elliptic curve (with modular invariant
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j mod p) by a torus.
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The second line contains three data concerning the reduction at p without
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any field extension.
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The first symbol describes the reduction at p of C. We use the symbols of
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Namikawa-Ueno for the type of the reduction (Namikawa, Ueno : "The complete
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classification of fibers in pencils of curves of genus two", Manuscripta
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Math., vol. 9, (1973), pages 143-186.) The reduction symbol is followed by
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the corresponding page number (or just an indiction) in the above article.
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The lower index is printed by { }, for instance, [I{2}-II-5] means [I_2-II-5].
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Note that if K and K' are Kodaira symbols for singular fibers of elliptic
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curves, [K-K'-m] and [K'-K-m] are the same type. Finally, [K-K'--1] (not the
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same as [K-K'-1]) is [K'-K-alpha] in the notation of Namikawa-Ueno. The figure
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[2I_0-m] in Namikawa-Ueno, page 159 must be denoted by [2I_0-(m+1)].
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The second datum is the group of connected components (over an algebraic
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closure of F_p) of the Neron model of J(C). The symbol (n) means the cyclic
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group with n elements. When n=0, (0) is
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the trivial group (1). H{n} is isomorphic to (2)x(2) if n is even and to
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(4) otherwise.
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Finally, f is the exponent of the conductor of J(C) at p.
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TEST EXAMPLES................................................
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1.
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Consider the curve defined by y^2=x^6+3*x^3+63.
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Run genus2reduction and enter 0 for Q(x), x^6+3*x^3+63 for P(x).
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Then you get :
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a minimal equation over Z[1/2] is :
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y^2 = x^6 + 3*x^3 + 63
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factorization of the minimal (away from 2) discriminant :
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[2, 8; 3, 25; 7, 2]
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p=2
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(potential) stable reduction : (V), j1+j2=0, j1*j2=0
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p=3
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(potential) stable reduction : (I)
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reduction at p : [III{9}] page 184, (3)^2, f=10
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p=7
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(potential) stable reduction : (V), j1+j2=0, j1*j2=0
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reduction at p : [I{0}-II-0}] page 159, (1), f=2
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the prime to 2 part of the conductor is 2893401
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It can be seen that at p=2, the reduction is [II-II-0] page 163, (1), f=8.
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So the conductor of J(C) is 2*2893401=5786802.
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2.
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Consider the modular curve X_1(13) defined by an equation
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y^2+(x^3-x^2-1)*y=x^2-x
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Run genus2reduction, and enter x^3-x^2-1 for Q(x) and x^2-x for
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P(x). Then you get
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a minimal equation over Z[1/2] is :
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y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
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factorization of the minimal (away from 2) discriminant :
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[13, 2]
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p=13
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(potential) stable reduction : (V), j1+j2=0, j1*j2=0
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reduction at p : [I{0}-II-0}] page 159, (1), f=2
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the conductor is 169
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So the curve has good reduction at 2. At p=13, the stable reduction is
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union of two elliptic curves, both of them have 0 as modular invariant.
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The reduction at 13 is of type [I_0-II-0] (see Namikawa-Ueno, op. cit,
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page 159). It is an elliptic curve with a cusp. The group of connected
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components of the Neron model of J(C) is trivial, and the exponent
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of the conductor of J(C) at 13 is f=2. The conductor of J(C) is 13^2.
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REMARKS..................................................................
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This program is based entirely on Pari (developed by C. Batut, D. Bernardi,
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H. Cohen and M. Olivier). For small primes 3, 5, 7, it has been tested at
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least twice for each type of reduction listed in Namikawa-Ueno (op. cit.).
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But it doesn't exclude bugs. Please report any problem or bug you could find
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to :
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[email protected]
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If you get this program by ftp, please send a message to the above
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address. You will be informed if there are further developments (especially
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concerning the reduction at p=2).
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Qing LIU
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CNRS, Laboratoire de Mathematiques Pures
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Universite de Bordeaux 1
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351, cours de la Liberation
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33405 Talence cedex
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FRANCE
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