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 that if E is an elliptic curve over Q, then there are infinitely many primes p for which the mod p reduction of E is supersingular. Later [El3] he extended his argument to prove the analogous statement for elliptic curves over number fields having a real place. But over other number fields the question is still open. In this note, we show that the analogous statement for Drinfeld modules over Fq(T ) is false: we exhibit Drinfeld modules having no primes of supersingular reduction. The obstruction is obtained from the mod ` representations associated to a Drinfeld module. The final section, which may be read independently of the rest of the paper, proves that such obstructions cannot exist for elliptic curves over number fields.