The Drinfeld modular Jacobian J(1)(n) has connected fibers.

Abstract

We study the integral model of the Drinfeld modular curve <italic>X</italic>₁(<italic>n</italic>) for a prime <italic>n</italic> &isin; <blkbd>Fq</blkbd> [<italic>T</italic>]. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod <italic>n</italic>. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order <italic>n</italic> in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of <italic>X</italic>₁(<italic>n</italic>) which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over <italic>n</italic>. Thus the mod <italic>n</italic> component group of <italic> J</italic>₁(<italic>n</italic>) is trivial, i.e. <italic>J</italic>₁(<italic>n</italic>) has connected fibers.