~ Computer Programs [an error occurred while processing this directive]

The Hecke Operators T_l and U_p acting on quaternionic modular forms

In his 2002 University of London PhD thesis, Daniel Jacobs proved results about the 3-adic slopes of the U_3 operator acting on certain spaces of automorphic forms defined over the definite quaternion algebra Q_2. He wrote gp-pari programs that computed these slopes.

His thesis contains one of these programs, and can be obtained from here. It's a nice introduction to the subject, as is Kevin Buzzard's paper On p-adic families of automorphic forms, which is available from his webpage.

I have been working on generalizing these programs to different spaces of automorphic forms and different U_p.

Version 0.1 of my generalized program

Here are the files for my program.

The generalized program which calculates the matrix of the U_p operator and its slopes

Auxiliary file which implements arithmetic functions on quaternions

Some code that computes the T_l operator (for l!=p)

There is also code to compute these automorphic forms with even smaller level structure at 2:

The generalized program which calculates the matrix of the U_p operator and its slopes

Auxiliary file which implements arithmetic functions on quaternions

More auxiliary programs

Functions to compute T_l for l!=p

This program runs under GP-Pari and works okay with version 2.2.9. It should work with other recent versions of Pari.

Here's an example run:

[Pari header]
? \r quaternionic_modular_forms.g
******************************
Program for calculating the action of U_p on quaternionic modular forms
Original written by Dan Jacobs, (c) 2002
Extended by L J P Kilford, (c) 2004-2005
******************************

Version 0.1

Verbose mode on

The prime p is 11
with extra level structure at 2
of the smaller level U_{sq}(11)

The p-adic accuracy is 40
The cosets are [[0, 10; 1, 0], [9, 0; 1, 5], [9, 0; 4, 5]]

Testing the factorisations ...
The factorisations are OK
Functions for testing matrices
Further factorisations
Finding the matrices for the generating functions
Defining the U_p operator
1
0
%27 = [2, 1, 1, 1, 1, 0]

Useful functions

Upmatrix(p_adic_accuracy,wt,dimension) - calculates the (dimension x dimension) matrix of the U_p operator acting on weight wt automorphic forms, calculated to p-adic precision p_adic_accuracy

charpowerseries(p_adic_accuracy,wt,dimension) - calculates the characteristic power series of Upmatrix(p_adic_accuracy,wt,dimension)

findSlopes(p_adic_accuracy,wt,dimension) - calculates the Newton slopes of charpowerseries(p_adic_accuracy,wt,dimension)

Tlmatrix(p_adic_accuracy,wt,dimension) - calculates the (dimension x dimension) matrix of the T_l operator acting on weight wt automorphic forms, calculated to p-adic precision p_adic_accuracy

charpowerseries_l(p_adic_accuracy,wt,dimension) - calculates the characteristic power series of Tlmatrix(p_adic_accuracy,wt,dimension)

findSlopes_l(p_adic_accuracy,wt,dimension) - calculates the Newton slopes of charpowerseries_l(p_adic_accuracy,wt,dimension)

Lloyd Kilford's home page
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