Number Theory Seminar

Classical modular curves associated with GL(2) are moduli spaces of elliptic curves with additional structure. While there are no Shimura varieties associated with the general linear group GL( r) for r >2,the situation is sharply different over function fields. The Drinfeld modular variety for GL( r) is the moduli space of Drinfeld modules of rank r (with auxiliary level structure). It is a smooth, affine scheme of relative dimension r−1. I will recall how various analogues of well-established tools in the classical context extend to Drinfeld modular varieties and their modular forms: the Hasse invariant, the Igusa tower, etc. I will then explain how to construct Hida families of Drinfeld modular forms and also mention what can be done for finite slope forms. Time allowing, I'll mention a classicality result of the type « small slope implies classical ». Joint work with G. Rosso.