Introduction to Drinfeld Modules

(1) Explicit class field theory for global function fields (just as torsion of Gm gives abelian extensions of Q, and torsion of CM elliptic curves gives abelian extension of imaginary quadratic fields). Here global function field means Fp(T ) or a finite extension. (2) Langlands conjectures for GLn over function fields (Drinfeld modular varieties play the role of Shimura varieties). (3) Modularity of elliptic curves over function fields: If E/Fp(T ) has split multiplicative reduction at ∞, then E is dominated by a Drinfeld modular curve. (4) Explicit construction of curves over finite fields with many points, as needed in coding theory, namely reductions of Drinfeld modular curves, which are easier to write equations for than the classical modular curves. Only the first of these will be treated in these notes.