Hida once described his theory of families of ordinary p-adic modular eigenforms as obtained from cutting “the clear surface out of the pitch-dark well too deep to see through” of the space of all elliptic modular forms. In this colloquium-style talk, we shall peer into the well of Drinfeld modular forms instead of classical modular forms. More precisely, we shall explain how to construct families of finite slope Drinfeld modular forms over Drinfeld modular varieties of any dimension. In the ordinary case (the “clear surface”), we show that the weight may vary p-adically in families of Drinfeld modular forms (a direct analogue of Hida’s Vertical Control Theorem). In the deeper & murkier waters of positive slope, the situation is more subtle: the weight may indeed vary continuously, but not analytically, thereby contrasting markedly with Coleman’s well-known p-adic theory.

Joint work with G. Rosso (Cambridge University / Concordia University (Montréal)).