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In this pages you will find a number of PARI-GP routines useful for number theory. This project is funded by a TARP grant.

There is a help file, which lists the help functions of the routines in this page and explains how to call them. To use it, download the file help.gp and read it by typing \r help.gp in a GP session.

The following routines allow the user to perform various calculations
on a quaternion algebra over the rationals. To use them download the
file
**
qalg.gp ** and then check
**
qalg.txt ** for details.

Here are some useful routines for making computations with positive
definite binary quadratic forms: computing representatives of class
group, class number, composition of forms, etc. To use it download
the file ** bforms.gp ** and
then check the file ** bforms.txt
** for details and examples.

This routine simply encodes the invariants of quartics that give a
Weierstrass equation for the Jacobian of such a curve according to
work of Weil. To use it download the file ** jac_quart.gp **.

(Preliminary version).
This routine compute a Weierstrass form for the Jacobian of a general
homogeneous cubic in 3 variables (over a field of arbitrary
characteristic); it is joint work with John Tate.
** jac_cubic.gp **.

(Even more preliminary version)
Polynomials giving a degree 9 map from the curve to its
Jacobian valid in any characteristic. To use download the above file
and also
** cubic-map.gp **
(** Warning: ** This last file is huge (1.6 megabytes)!)

Computation of Dedekind's eta function on CM points using its modular
properties to relate the value to that of the corresponding point in
the standard fundamental domain. To use it download the file **
etafunction.gp **.

Computes
the matrix of change of variables taking a skew-symmetric matrix to
its standard symplectic form. To use it download the file
** skewsymbas.gp **.

The following routines calculate the Igusa invariants of a sextic
(after Mestre). To use them
download the file
**
inv.gp **

The following routines calculate elliptic units associated to an arbitrary
order in an imaginary quadratic field. Based on
a paper by Farshid Hajir and Fernando Rodriguez-Villegas. To use them
download the file
**
expell.gp **
. For more information about the routines, and to see some examples, download the file
**
expell.txt **
You will need the files
** bforms.gp **
and
** etafunction.gp **.

The following routine calculates the p-adic Gamma function using an
expansion due to Dwork. To use it download the file
**
gammap.gp **

These are some elementary routines, complementary to those of pari, for dealing with several variables polynomials. To use it download the file
**
polynomials.gp **

Given a non-singular symmetric 3 by 3 matrix with rational entries
this routine returns a list of primes p for which the corresponding
conic has no non-trivial solution over the p-adics.
To use download
**
conic.gp **

This implementation of Pollard's p-1 factoring method was written by
F. Voloch for his
**
Applied Number Theory** course at UT. To use download
**
pollard.gp **

This file contains some routines for doing arithmetic in quaternion algebras,
specifically for computing the Brandt matrices in a quaternion algebra
ramified at a prime p and infinity.
To use download the file: ** qalgmodforms.gp **.
You will also need the files: ** qalg.gp **
, ** bforms.gp ** and **
qforms.gp **.

There is also the help file **
qalgmodforms.txt ** you can download which gives a brief explation
of the routines, and has some examples.

Given a matrix in Sl_n (Z/NZ), computes a lift to Sl_n(Z), i.e. a matrix with
determinant 1, such that reduces to the given matrix modulo N. Actually it
works if the original matrix has determinant -1. To use, download the file
** liftslg.gp **

These are some elementary routines that work for searching elements in a vector
, ordering elements, making permutations, and some other basic things. It also
has some routines for eliminating a column or a row of a matrix. To use it, download the file ** vectors.gp **

**Ariel Pacetti**

** Address: **Department of Math, UT Austin, Austin, TX
78712

**
Phone: **
(512) 475-8688

**
Office: **
RLM 12.166

**
Fax: **
512-471-9038

**
E-mail: **
apacetti@math.utexas.edu

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Send questions, comments to villegas@math.utexas.edu or apacetti@math.utexas.edu .

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