LOCAL HEIGHT FUNCTIONS AND THE MORDELL-WEIL THEOREM FOR DRINFELD MODULES

We prove an analogue for Drinfeld modules of the Mordell-Weil theorem on abelian varieties over number fields. Specifically, we show that if phi is a Drinfeld A-module over a finite extension L of the fraction field of A, then L considered as an A-module via phi is the direct sum of a free A-module of rank aleph(0) with a finite torsion module. The main tool is the canonical global height function defined by Denis. By developing canonical local height functions, we are also able to show that if phi is defined over the ring of S-integers O-s in L, then O-s and L/O-s considered as A-modules via phi also are each isomorphic to the direct sum of a free A-module of rank aleph(0) with a finite torsion module. If M is a nontrivial finite separable extension of L, then the quotient module M/L as well is isomorphic to the direct sum of a free A-module of rank aleph 0 with a finite torsion module. Finally, the original result holds if L is replaced by its perfection.