Average Distribution of Supersingular Drinfeld Modules

Drinfeld Modules with No Supersingular Primes

We give examples of Drinfeld modules φ of rank 2 and higher over Fq(T ) that have no primes of supersingular reduction. The idea is to construct φ so that the associated mod ` representations are incompatible with the existence of supersingular primes. We also answer a question of Elkies by proving that such obstructions cannot exist for elliptic curves over number fields. Elkies [El1] proved t...

Hecke Modules and Supersingular

Let F be a nonarchimedean local field of odd residual characteristic p. We classify finite-dimensional simple right modules for the pro-pIwahori-Hecke algebra HC(G, I(1)), where G is the unramified unitary group U(2, 1)(E/F ) in three variables. Using this description when C = Fp, we define supersingular Hecke modules and show that the functor of I(1)-invariants induces a bijection between irre...

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) Modular...

The arithmetic of Drinfeld modules

The arithmetic of Drinfeld modules

Drinfeld Modules and Elliptic Sheaves