Hecke Operators

The computations are done essentially on the cohomology of \(\Gamma \backslash \bf{H}_3\), and so the for a non-principal ideal \(P\), the Hecke operator \(T_P\) does not act on this space, but sometimes the Hecke action can be deduced from the action of principal Hecke operators. See for instance [Lingham, 2005]. Specifically, suppose \({\mathfrak{p}}\) is a prime ideal that is prime to the level. There exists an ideal \({\mathfrak{a}}\), prime to \({\mathfrak{p}}\) and the level, such that \({\mathfrak{a}}^2 {\mathfrak{p}}\) is principal. Then the composition \(T_{{\mathfrak{a}},{\mathfrak{a}}}T_{{\mathfrak{p}}}\) acts on the cohomology.

HeckeOperator(M, P): ModFrmBianchi, RngOrdIdl -> Mtrx

This returns a matrix representing a certain Hecke action \(T\) on the space \(M\) of Bianchi modular forms, with respect to the fixed basis of \(M\). The ideal \(P\) must be principal (but not necessarily prime) or prime and a square in the class group. The ideal must also be coprime to the level (except if the space is dimension 0). If \(P\) is principal, then \(T\) is the Hecke operator \(T_P\). When \(P\) is prime and a square in the class group, \(T\) is the composition \(T_{{\mathfrak{a}},{\mathfrak{a}}}T_P\) for a suitably chosen ideal \({\mathfrak{a}}\).

Example: hecke (ex-72d295)

We continue the previous example.

> _<x> := PolynomialRing(Rationals());
>  F := NumberField(x^2+14);
>  OF := Integers(F);
>  level := (Factorization(3*OF)[1][1])^2;
>  M9 := BianchiCuspForms(F, level);
>  P:=Factorization(23*OF);
>  P[1,1];
Prime Ideal of OF
Two element generators:
    [23, 0]
    [3, 1]
>  HeckeOperator(M9, P[1,1]);
[8]
>  P[2,1];
Prime Ideal of OF
Two element generators:
    [23, 0]
    [20, 1]
>  HeckeOperator(M9, P[2,1]);
[-8]
>  HeckeOperator(M9, 2*OF);
[1]

Run in calculatorCopy input

Since this cuspidal space has dimension \(1\), it consists of a single eigenform, whose eigenvalues can be read from the Hecke matrices.