Average Distribution of Supersingular Drinfeld Modules Original Research Article

Letφbe a rank 2 Drinfeld module defined over imageq(T). For each monic prime polynomialpset membership, variantimageq(T) which is a regular prime ofφ, the reduction ofφatpis a rank 2 Drinfeld moduleφpover the finite field imageq(T)/(p); depending on the structure of the ring End(φp), the regular primepis either a supersingular or an ordinary prime ofφ. We prove in this paper that,on average, supersingular primes are distributed according to the Lang-Trotter conjecture (for Drinfeld modules). We first show this result averaging over all Drinfeld modules, and then over all isomorphism classes of Drinfeld modules.