David R. Hayes

The Arithmetic of function fields : proceedings of the workshop at the Ohio State University, June 17-26, 1991

These proceedings (workshop held at Ohio State U., June 1991) introduce new techniques which center on Drinfield modules and their higher dimensional analogs, t-modules. The theory of Drinfield modules has grown rapidly over the last few years, yet is unknown to most mathematicians. This volume intr

Ramifications Arising from Drinfeld Modules

In this paper, we study various ramifications arising from division points of Drinfeld modules, abelian T -modules, formal modules, etc.. A motivation for this is to know how many isogeny classes and isomorphism classes of Drinfeld A-modules exist over a finite extension of the fraction field of A . We will see ( cf. Remark (3.4) ) that, modulo the isogeny conjecture, an isogeny class can contain infinitely many isomorphism classes and, without any restriction on ramification at the infinite places, there can be infinitely many isogeny classes. To explane some of the results, let F be a function field in one variable over a finite field, ∞ a fixed place of F , A the ring of elements of F which are regular outside ∞ , and K a finite extension of F . Given a Drinfeld A -module φ over K and a prime v of A , we denote by K(φ; v) the field of v -division points of φ . Then it turns out ( Corollary 1.6 ) that the ramification at various primes in the tower (K(φ; v)/K)n≥1 is bounded at the places over ∞ by a divisor depending only on φ , and at the finite places, it is controlled in a fairly precise way in terms of the “discriminant” ∆(φ). Roughly speaking, ∆(φ) is the coefficient of the leading term of the defining equation of φ . For finite places, this result is analogous to the case of abelian varieties over number fields. ( At least one has the HermiteMinkovski theorem for number fields, which assures the existence of an estimate of discriminants. ) But at infinite places, there occur new phenomena, which we describe by example in §2. We construct explicitely an infinite family of Drinfeld modules with everywhere good reduction and with ramification at infinity becoming arbitrarily large ( Example 2.1 ), as well as an infinite family of mutually nonisomorphic Drinfeld modules with everywhere good reduction and with bounded ramification at infinity ( Example 2.2 ). In §3, we give a proposition on v -adic Galois representations ( a positive characteristic version of a theorem of Faltings ), and discuss how many isomorphism and isogeny classes can exist. §4 and §5 are generalizations of §1 to the cases of finite submodules of higher dimensional formal modules. Theorem (4.6) is an A -module version of Théorème 1 of [5].