Hecke ring definition

[CBirkbeck]

• Define Hecke rings abstractly like Shimura does.
• Define spaces of modular forms of weight k and any level in a commensurability class. Can try two different approaches. First define IsModularForm predicate on function on the upper half plane. Then just take the union. Alternatively, define the direct limit of the spaces we have now and deal with the pain. (See Automorphic forms and representations - Bump ).
• Define Eigenforms
• Define maps between modular forms of different levels i.e. if f has level N then f(d z) has level Nd.
• Define Newforms, Primitive forms.