Automorphic Form - Formulation

Formulation

The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when j is derived from a Jacobian matrix, by means of the chain rule.

In the general setting, then, an automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

1. to transform under translation by elements according to the given automorphy factor j;
2. to be an eigenfunction of certain Casimir operators on G; and
3. to satisfy some conditions on growth at infinity.

It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for . In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have F as eigenfunction; this ensures that F has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where G/Γ is not compact but has cusps.