Average Distribution of Supersingular Drinfeld Modules

✍ Scribed by Chantal David

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 657 KB
• Volume: 56
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0025

✦ Synopsis

Let , be a rank 2 Drinfeld module defined over F q (T ). For each monic prime polynomial p # F q (T ) which is a regular prime of ,, the reduction of , at p is a rank 2 Drinfeld module , p over the finite field F q (T )Â( p); depending on the structure of the ring End(, p ), the regular prime p is either a supersingular or an ordinary prime of ,. 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.

1996 Academic Press, Inc. of elliptic curves. Brown [2] was the first to examine this new situation, and to notice that most of the theory extends from elliptic curves to Drinfeld modules. Then, similarly to the classical case of elliptic curves (see [6] and [14]), he obtained lower bounds for the number of supersingular article no.