Computational Number Theory

## Computational Number Theory PAGE UNDER CONSTRUCTION

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.

• Quaternion algebras

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.

• Jacobian of y^2=f(x) with f of degree 4

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 .

• Jacobian of plane cubics

(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)!)

• Dedekind's eta function

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 .

• Skew-symmetric matrix

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 .

• Igusa invariants

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

• Explicit elliptic units

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

• Several variables polynomials

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

• Conics

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

• Pollard method

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

• Brandt Matrices

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.

• Lifting matrices

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

• Elementary vectors routines

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

Fernando Rodriguez Villegas
Address: Department of Math, UT Austin, Austin, TX 78712
Phone: (512) 471-1137
Office: RLM 9.164
Fax: 512-471-9038
E-mail: villegas@math.utexas.edu

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

Last updated [an error occurred while processing this directive]
Send questions, comments to villegas@math.utexas.edu or apacetti@math.utexas.edu .