View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Number Theory 103 (2003) 234–256 http://www.elsevier.com/locate/jnt Formal Drinfeld modules Michael Rosen Department of Mathematics, Brown University, Providence, RI 02912, USA Received 24 June 2002; revised 15 February 2003 Communicated by K.A. Ribet Abstract The formal group of an ellipticcurveat a finite prime of the field of definition has proven to be a useful tool in studying the ellipticcurve.Moreover, these formal groups are interesting in themselves. In this paper we define and study formal Drinfeld modules in a general setting. We also define the formal Drinfeld module associated to a Drinfeld module at a finite prime. The results are applied to the uniform boundedness conjecture for Drinfeld modules. r 2003 Elsevier Inc. All rights reserved. Let E be an elliptic curve defined over a local field K; and let R be the ring of integers in K: It is well known that one can associate to E a formal group defined over R: This formal group is a useful tool for studying the kernel of reduction modulo the maximal ideal of R; the points of finite order on E; etc. In this paper we will take K to be a local field of positive characteristic p40: We will define the notion of a formal Drinfeld R module and relate it to the standard definition of a Drinfeld module. This idea turns out to be a special case of the notion of a formal R module which goes back to Lubin and Tate [12]. In another direction it is also a special case of the notion of a formal t-module due to Anderson [1].Nevertheless, most of the material we present is not encompassed by these earlier works. These formal Drinfeld modules arise naturally in the following way. Let k=F be a global function field over a finite field F: Let N be a prime of k and ACk the subring of functions whose only poles are at N: Let r be a Drinfeld module on A defined over some finite extension E of k; and let w be a prime of E lying over a finite prime vaN of k: Finally, we assume that r has stable reduction at w: We will show how to define the formal completion of r at w; which will be a formal Drinfeld Rv module, where Rv is the completion of the valuation ring of v: This Rv module will be defined E-mail address: michael [email protected]. 0022-314X/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00111-2 ARTICLE IN PRESS M. Rosen / Journal of Number Theory 103 (2003) 234–256 235 over the completion of the valuation ring of w: As in the case of elliptic curves, this formal Drinfeld module will give information about the kernel of the reduction map at w and about torsion points for r: This paper is organized as follows. In Section 1 we will define formal Drinfeld modules and explain the relationship with formal R modules as developed, for example, in Section 1 of Drinfeld [3] or Chapter IV of Hazewinkel [9]. We then discuss the category of formal Drinfeld modules. Let R be a complete discrete valuation ring with finite residue class field. In Section 2 we will show how a formal Drinfeld R module allows us to make the maximal ideal in a local R algebra into an R module in a new way. We explore this structure by means of the logarithm and exponential maps, which convert the new R module structure into the standard R module structure and vice versa. The domains of convergence of these maps are also determined. In Section 3 we state and prove a Weierstrass preparation theorem for the non-commutative ‘‘twisted’’ formal power series ring that is at the base of all of our considerations. A more general version of this result is due to Honda [10]. This is used to determine the structure of the torsion points on a formal Drinfeld module. In Section 4, we start with a Drinfeld module r on an affine subring of a global function field and define the formal completion of r at a finite prime. Applications to reduction mod P and to the structure of torsion points are given. In the last section, Section 5, we describe the uniform boundedness conjecture in the context of Drinfeld modules, see [13]. Using formal Drinfeld modules, we give a version of Poonen’s proof of his theorem, which states the conjecture is true for rank 1 Drinfeld modules. We also give some weaker, but still interesting, results for Drinfeld modules of rank 2 or greater. 1. Let K be a locally compact field of characteristic p40: It is well known that such a field has a rank one valuation v and is complete in the corresponding topology. The ring of integers R ¼faAK j vðaÞX0g is a discrete valuation ring. Let PCR be its maximal ideal and F ¼ R=P the residue class field. One can show that R contains a subfield isomorphicto F: We call this subfield F as well. Let p be a uniformizing parameter, i.e. P ¼ðpÞ: Then one can prove that R ¼ F½½p; the ring of formal power series in p; and K ¼ FððpÞÞ; the field of formal Laurent series in p: We fix a subfield FoDF with q elements and define d ¼ deg P ¼½R=P : Fo: Let L be a commutative R algebra, and let i : R-L be the structure map. We defineP the twisted power series ring Lfftgg to be the ring whose elements are power N n A series n¼0 cnt with cn L for all n: Two monomials multiply according to the rule n btnctm ¼ bcq tmþn; and this multiplication extends uniquely to a multiplication on Lfftgg in the usual way. Let D : Lfftgg-L be the map that assigns to a power series its constant term. D is a homomorphism and its kernel is the two-sided ideal ðtÞ: ARTICLE IN PRESS 236 M. Rosen / Journal of Number Theory 103 (2003) 234–256 Definition. A formal Drinfeld R module defined over L is a ring homomorphism - a r : R Lfftgg with three properties: DðraÞ¼iðaÞ for all aAR; rðRÞgL; and rp 0 for one (and thus all) uniformizing parameters p: This definition is very much like the definition of a Drinfeld module (see [7,8] or [14]) except that in a Drinfeld module the ring R is a ring of functions on an affine curve over a finite field, and one uses a twisted polynomial ring instead of a twisted power series ring. We will sometimes use the notation Dˆ RðLÞ to denote the set of formal Drinfeld R modules defined over L: We briefly recall the definition of a formal R module, and show how the above notion of a formal Drinfeld module arises as a special case. Let R be a commutative ring and B a commutative R algebra with i : R-B the structure morphism. Let FðX; YÞAB½½X; Y be a formal group in one variable defined over B: We say that F is a formal R module if there is a homomorphism r : R-EndBðFÞ such that for all 2 aAR we have raðXÞiðaÞX ðmod X Þ: Now, suppose B has characteristic p40; and ˆ take for F the additive formal group over B defined by G aðX; YÞ¼PX þ Y: Then A ˆ N pn gðXÞ B½½X is in EndBðG a=BÞ if and only if gðXÞ is of the form n¼1 bnX : It ˆ follows easily that EndBðG a=BÞ is isomorphicto Bfftgg: The commutation rule is tb ¼ bpt: It is now clear how our definition fits into this general framework. For more details on the general case, see [3] and, especially, [9]. For the rest of this paper we assume that L is an integral domain. It is an easy consequence that Lfftgg has no zero divisors. Lemma 1.1. If r is a formal Drinfeld R module over L; then r is a monomorphism. a Proof. This is clear since r is a homomorphism and rp 0: & P N n For a non-zero power series f ¼ n¼0 cnt ; define ordtð f Þ to be the smallest subscript n such that cna0: Note that ordtð f Þ¼n if and only if f can be written as n ht ; where DðhÞa0: Also, ordtð fgÞ¼ordtð f ÞþordtðgÞ: 0 Let pAR be a uniformizing parameter and consider rp: The integer h ¼ ordtðrpÞ 0 will play an important role. Since DðrpÞ¼iðpÞ; we must have h ¼ 0ifi is a monomorphism. If i is not a monomorphism, then kerðiÞ¼P; and so h040: 0 Lemma 1.2. The integer h ¼ ordtðrpÞ is divisible by d ¼ deg P: 0 0 h0 Proof.P If h ¼ 0; there is nothing to prove, so assume h 40: Then rp ¼ f t ; where n h0 f ¼ cnt and c0a0: Let aAR: Comparing the coefficients of t on both sides of the equality rarp ¼ rpra; we find 0 qh iðaÞc0 ¼ c0iðaÞ : Since c0a0 and L is an integral domain, this shows that every element of iðRÞDF h0 satisfies xq ¼ x: Since #ðFÞ¼qd ; it follows that djh0: & ARTICLE IN PRESS M. Rosen / Journal of Number Theory 103 (2003) 234–256 237 Definition. Let r be a formal Drinfeld R module defined over L: Let pAR be a 0 0 uniformizing parameter and h ¼ ordtðrpÞ: If h ¼ 0; we say that the height of r is 0: If h040; we define the height of r to be h0=d: We denote the height of r by htðrÞ: It is almost immediate that this definition is independent of the choice of uniformizing parameter.