MAGMA package HeckeAlgebra

Summary

It is of some interest to know the structure of the Hecke algebra of mod p modular forms. For example, there are cases where it is not known whether the Hecke algebra is a Gorenstein ring.

This package performs the Hecke algebra computation on mod p modular symbols in quite an efficient way, building on William Stein's implementation of modular symbols.

The package contains functions for the computation of Hecke algebras of modular forms over finite fields of characteristic p in the following cases:

• all eigenforms for a given level N, weight k and Dirichlet character of modulus N,
• dihedral modular forms coming from the Hilbert class field of a quadratic number field,
• icosahedral modular forms (currently only in characteristic p=2) given by an A5-polynomial.

With Lloyd Kilford we have used this package for computations concerning the Gorenstein property of Hecke algebras of modular forms of prime weights and written the article On the failure of the Gorenstein property for Hecke algebras of prime weight.

Here is a table summarising our computations. The complete data in tar.gz.

Download

Here is the complete code, as well as the documentation and an example as a tar.gz-file.

Here is the manual. It contains a note on the installation, a long example and precise descriptions of the functions included in the package.

Old version

This old version is the one referred to in the article with Kilford, which is the only reason why the code is still here.

HeckeAlgebra.mg (MAGMA source).

Example.mg (an example source).

Manual.pdf (the manual).

Last modification: 8 August 2008.