Hecke operators for unitary groups over imaginary quadratic number fields

Introduction

This repository contains the code that was used to compute the Hecke operators presented in the article Automorphic Forms on Feit's Hermitian Lattices (N. Dummigan and S. Schönnenbeck, 2019, preprint). The implementation depends on the 'forms'-package of Markus Kirschmer's which can be found on his homepage (see 'Magma-packages'). Anybody is welcome to use these algorithms for their own research as long as the relevant articles are cited.

Usage

Overall the usability is subpar since most of the algorithms were written for one time use. Nevertheless, we believe them to be of general interest. Use at your own risk.

Start any computation by attaching the 'forms'-package:

    AttachSpec("path/to/lat.spec");

Constructing Feit's unimodular lattices

To construct the 20 unimodular lattices from Feit's original article ("Some lattices over Q(sqrt(-3))", W. Feit, J. Alg., 1978) simply call

    load "path/to/Magma/FeitLattices.m";

This will construct a list 'Unimod12' containing the Hermitian lattices in question in the exact order we consider in our article. In addition some intermediate results are constructed, e.g. you will get lists

    Lat6, Lat8, Lat9, Lat10, and Lat11

containing the indecomposable unimodular lattices in dimension 6, 8, 9, 10, and 11, respecitvely.

Computing Hecke operators

For computing Hecke operators call

    load "path/to/Magma/Hecke.m";

This provides you with two functions

    IntertwiningOperator(GenusOfL, P);
    HeckeOperator(L,P);

The first one takes as argument a list of representatives of a genus of lattices and an ideal P and computes the intertwining operator S_P (see Proposition (3.5) of the article) as well as a list of representatives of the corresponding secondary genus of lattices. These can be used in a straightforward manner to compute the Hecke operator of the genus at the prime P. However, we also provide a convenience function 'HeckeOperator' that takes only a lattice L and a prime P and computes (in this order):

• The neighbouring operator of the genus of L at P
• The intertwining operator S_P
• The corresponding operator S_P'
• Representatives for the genus of L
• Representatives for the corresponding secondary genus

It is the user's responsibility to check that the conditions of Proposition (3.5) are fulfilled when using these functions as they will also run in other cases in which the results cannot be trusted.

The function 'HeckeOperator(L,P)' can be made quite a bit faster if we already know as system of representatives of the genus of L (or the second genus) and the corresponding automorphism groups. It should be relatively easy to adjust the code accordingly.

Bugs

It is highly likely that these algorithms contain bugs. In case you find one please feel free to let me know or - if you manage to fix it yourself - to submit a pull request.