-------------------------------------------------------------------
Time-stamp: <2000-05-12 18:56:57 apacetti>

Description: Explicit Elliptic Units

Based on "Explicit Elliptic Units I" by Farshid Hajir and Fernando
Rodriguez-Villegas Duke Math. J. 90, (1997) 495-521

Authors:   Fernando Rodriguez-Villegas
	   villegas@math.utexas.edu
           University of Texas at Austin

	   Ariel Martin Pacetti	 
	   apacetti@math.utexas.edu
	   University of texas at Austin
-------------------------------------------------------------------

This file describes the PARI-GP routines in the file expell.gp, which
is actually based on the paper "Explicit Elliptic Units I" by Fernando
Rodriguez-Villegas and Farshid Hajir, Duke Math. J 90, (1997) 495-521.

To use: in a GP session issue the command

\r expell.gp

(if that's the name you gave the file; you might need more of its path
if you GP session is running with another default directory).

For using this program, you need to download the files "bforms.gp",
"etafunction.gp". The file bforms.gp works with quadratic forms,
see the file bforms.txt for more information. The file etafunction.gp
contains an algorithm for computing the eta of a form [a,b,c].

Some notations we use on the routines is call with the letters:

    d: the discriminant.
    q: a quadratic form.
    w: nuber of roots of unity in the ring class field of disc. -d.
    e:

There are some elementary auxiliar routines, like sqfp, to compute the
square free part of an integer, for example:

     sqfp(2^3*3^2*5)

will output:

     10

Or rd, which round the coefficients of a polynomial if they're closed
to an integers. For example:

     rd(1.0000049*x+2)

will output;

     x+2

The main routines are:

    * wx: which computes w_H= number of roots of unity in H= ring
class field of discriminant d (where d<0). For example:

    wx(-23)

which will output:

    2

    * wz: which computes w_K= number of roots of unity in K itself (i.e. only
interesting if K=\Q(i), \Q(\sqrt{-3})). For example:

    wz(-1)

which will output:

    4

or: wz(-23)

which will output:

    2

    * nc: computes for each unit u_C, the smaller number n such that
u_C^n is in H. For example:

    nc(-23)

which will output:

    [1, 1, 1]

or for example:   nc(-39)

which will output:

    [1, 3, 3, 1]

    * elluts: computes the minimal polynomial of u_C^n_C (i.e. the
smallest power such that lies in H). The answer is
a vector [n_C,pol_C]. For example:

    elluts(-23)

which will output:

    [[1, x^3 - x + 1]]

or:    elluts(-39)

which will output:

    [[3, x^4 + 3*x^3 - 4*x^2 + 2*x - 1], [1, x^4 - x^3 - x^2 - x + 1]]

    * ellnum: given a degree 0 divisor of binary forms, i.e. [
...[n_Q,Q]...] computes the minimal polynomial of the smallest power
of "prod{eta(Q)^n_Q}" which is in H. For example:

    ellnum([[1,[5,1,2]],[-1,[11,37,32]]])

which will output:

    [-39, 6, 3, 1, [55, 81, 30], 4, x^8 - 0.7603305785123966942148681982*x^7 + 0.02035380096987910661839066615*x^6 - 0.1098872689114289601091881432*x^5 + 0.1489526855665942760729368886*x^4 - 0.02270398118004730580768350065*x^3 + 0.0008688699956406284841536893919*x^2 - 0.006706043590514917830719526786*x + 0.001822294453944271149652064099]


    * sput: given a binary form q, there is a special unit associates
to it, corresponding to the degree 0 divisor
v=[[2,Q],[-1,Q*Q],[-1,Q_0]], where Q*Q is the product in the class
group, and Q_0 is the principal form. The answer is a vector:
[d,w,e,r,pol], where r is the power of \eta actually used, and pol is
the minimal polynomial of the unit. The first coefficient of the form
must be coprime to 6*d. For example:

    sput([1, 1, 6])

which will output:

    [-23, 2, 2, 1, x^3 - 3*x^2 + 3*x - 1] 

or:    sput([13, 17, 6])

which will output:

    [-23, 2, 2, 1, x^3 + x^2 - 1]


    * info: gives some information about a divisor, needed for another
routines as sput and ellnum. The output is a vector of the form:
[d,w,e,m,q], where d,w,e are as usual and m,q represent the conductor
of the quadratic symbol associated to the divisor described as: m\A, m
\in \Z and \A is the primitive ideal corresponding to q. For example:


    info([[1,[5,1,2]],[-1,[11,37,32]]])

which will output:

    [-39, 6, 3, 1, [55, 81, 30]]


    * et: given a binary form q, computes \eta(\A) as defined in
EEU-I; q is a binary form corresponding to \A and w is the value 
of \tilde{w} (given as an optional input to function to avoid
recomputing). For example:

    et([1, 1, 6])

which will output:

    0.4931478831501138857614502365 + 0.2042685414563594868572526134*I

or:    et([13, 17, 6])

which will output:

     -0.7844318209809436042191571769 + 0.8634485175044292015100576765*I