Formal Drinfeld Modules

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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].

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. We also note that this deﬁnition is closely related to the notion of the height of a homomorphism between formal R modules given in Section 1of[4].

Lemma 1.3. Let r be a formal Drinfeld R module over L: Then for all non-zero aAR; we have ordtðraÞ¼htðrÞÁd Á vðaÞ:

n A n Proof. Let a ¼ up ; where u R is a unit and n ¼ vðaÞ: Then ra ¼ rurp; which 0 implies ordtðraÞ¼ordtðruÞþn ordtðrpÞ¼vðaÞh ¼ h Á d Á vðaÞ: &

0 0 Deﬁnition. Let r; r ADˆ RðLÞ: A morphism from r to r is an element g of Lfftgg 0 0 such that gra ¼ ra g for all aAR: We denote by HomLðr; r Þ the set of such morphisms.

0 One sees that HomLðr; r Þ is an abelian group under addition of twisted power 0 00 0 00 series. Moreover, the map HomLðr ; r ÞÂHomLðr; r Þ-HomLðr; r Þ given by ðg; f Þ-gf is bi-additive. One easily checks that these deﬁnitions make Dˆ RðLÞ into a category. Note that EndLðrÞ :¼ HomLðr; rÞ is a ring. In fact, it is a subring of Lfftgg: Also note that rðRÞ is in the center of EndLðrÞ:

Proposition 1.4. If r is a Drinfeld module of height 0; then the map 0 D :HomLðr; r Þ-L is a monomorphism.

0 Proof. Let g be a non-zero element of HomLðr; r Þ: Suppose DðgÞ¼0: Then g ¼ n nþ1 cnt þ cnþ1t þ ?; where cna0 and n40: Let aAR and compare the coefficient of n 0 qn t on both sides of gra ¼ r ag: We find that cniðaÞ ¼ iðaÞcn; which implies that iðRÞ is a finite field. Since r has height 0; we must have iðpÞa0; and so i is one-to-one. Since R is infinite, we cannot have that iðRÞ is finite. Thus DðgÞa0; and so D is a monomorphism. &

Corollary. If the height of r is 0; then EndRðLÞ is a commutative ring.

Proof. The map of EndLðrÞ to L given by f -Dð f Þ is a ring monomorphism. &

As might be expected, a non-trivial morphism is height preserving. This is the content of the following lemma, which follows easily from the deﬁnitions.

0 0 Lemma 1.5. If HomLðr; r Það0Þ then htðrÞ¼htðr Þ: ARTICLE IN PRESS

238 M. Rosen / Journal of Number Theory 103 (2003) 234–256

We will now specialize L somewhat. Let L be a finite algebraicextension of K and O the integral closure of R in L: O is a discrete valuation ring. Let M ¼ðPÞ be its maximal ideal and E ¼ O=M its residue class field. Let w be the discrete valuation of L normalized by the condition wðPÞ¼1: The number e ¼ wðpÞ is the ramification index of the extension L=K: We will consider formal Drinfeld R modules over O: The structure morphism is just the injection of R into O: Let r be such a formal module. There is a well-defined homomorphism, reduction modulo M; from Offtgg-Efftgg: We denote this map % by a bar, f -f: Then a-r% a ‘‘almost’’ defines a formal Drinfeld R module over E with structure morphism R-R=P-O=M ¼ E: It will certainly do so if we require that % r% pa0:

Deﬁnition. Let rADˆ RðOÞ: We say r has stable reduction if r% : R-Efftgg is a formal Drinfeld R module over E:

0 0 0 Proposition 1.6. Let r; r ADˆ RðOÞ; and let HomOðr; r Þ-HomEðr%; r% Þ be the homomorphism given by reduction modulo M: If r has stable reduction, then this map is a monomorphism (compare [11] where a similar result is proven in the context of formal groups).

0 % % m Proof. Suppose f AHomOðr; r Þ is such that f a0; but f ¼ 0: We can write f ¼ P g where m40 and g%a0%: 0 m 0 m From f rp ¼ rp f ; we deduce P grp ¼ rp P g: We now calculate ! XN 0 m n m rp P ¼ p þ cnt P n¼1 XN m mqn n m ¼ pP þ cnP t ¼ P h; n¼1

% % m m where h ¼ 0: Substituting, we ﬁnd P grp ¼ P hg; and so grp ¼ hg: Reducing % % % modulo M; we have g%r% p ¼ hg% ¼ 0: Since r has stable reduction we know that r% pa0: It follows that g% ¼ 0%: This is a contradiction, which establishes the proposition. &

In light of the above proposition it is interesting to investigate formal Drinfeld R modules defined over a finite field E: Since this is off to the side of our main considerations, we just sketch how this process goes. If i : R-E is the structure morphism, we must have that kerðiÞ¼P: Thus, i induces a monomorphism of F ¼ R=P-E: Recall that R contains a copy of F; which we continue to call F; and we identify it with its image under i: Let n ¼½E : Fo and n F ¼ t : Suppose that c is a formal Drinfeld R module over E: Then, clearly, Fo½½F is o # o in the center of G ¼ EndEðcÞ: Set G ¼ G Fo½½FFoððFÞÞ: G is a division algebra. Since c imbeds R into G; this extends uniquely to an imbedding of K into Go: We identify K with its image under this embedding. ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 239

Theorem 1.7. With the above notations, the center of Go is KðFÞ: As a central division algebra over the local ﬁeld KðFÞ; the invariant of Go is ½KðFÞ : K=h: The number s ¼ h=½KðFÞ : K is an integer, and ½Go : KðFÞ ¼ s2: Finally, G is the maximal R-order in Go:

We will not prove this theorem here. We note the strong similarity to a result, Proposition 2.1 of Drinfeld [4], which concerns the endomorphism rings of Drinfeld modules over a finite field. See also [2,6]. We end this section with two remarks about the existence of formal Drinfeld modules. Perhaps the major effort in Section 1 of [3] and Section 21 of [9] is devoted to producing a universal formal R module. This is a formal R module FR over a ring LR such that if B is an R algebra and F a formal R module over B; then F is obtained from FR by applying a homomorphism from LR to B to the coefficients of FR and the endomorphisms of FR giving the R action. When R is the ring of integers in a local field, Drinfeld proves that LR is a graded polynomial ring over R in denumerably many variables. See Proposition 1.4 in [3]. Applied to formal Drinfeld modules, this proposition shows that they exist in abundance. In Section 4 we will show how to construct a formal Drinfeld module associated with a Drinfeld module having stable reduction at a given prime of the field of definition. This will supply us with a large collection of naturally occurring formal Drinfeld modules. Another large collection of formal Drinfeld modules can be constructed in the special case that F ¼ Fo: In this case R ¼ Fo½½p; where p is a uniformizing parameter of R: The non-commutative ring Offtgg is a local ring. The maximal ideal of O can be characterized by M ¼ff AOfftgg j Dð f ÞAMg: More generally, Mn is the set of n Tpower series whose first n coefficients are in M : From this, one sees concretely that n n M ¼ð0Þ; and that Offtgg is complete in the maximal ideal topology.P Let f be %a% A n any element in this ring withP Dð f Þ¼p and f 0: If a R; then a ¼ n anp with A n - an Fo for all n: Set ra ¼ n an f : Then a ra is well defined, and is a formal Drinfeld R module defined over O (one needs the fact that f commutes with every element of Fo). This construction is analogous to the construction of Drinfeld modules over a polynomial ring, F½T:

ˆ 2. Let rAD RðOÞ: Using r we can make M; the maximalP ideal of O; into an R n module by deﬁning a%m ¼ raðmÞ: More explicitly, if ra ¼ n bnt then XN qn a%m ¼ bnm : n¼0 This series is easily seen to converge to an element of M: We will use the notation Mr to denote M considered as an R module via this new action. It turns out that we can map Mr to L (i.e. L with the standard action of R)bya logarithm map. This follows from a very general construction in the theory of formal ARTICLE IN PRESS

240 M. Rosen / Journal of Number Theory 103 (2003) 234–256

R modules, see the discussion in 21.5.7 of [9]. We present a more elementary treatment which will be useful in determining domains of convergence.

Proposition 2.1. Let rADˆ RðOÞ: There is a uniquely deﬁned lrALfftgg with DðlrÞ¼1 such that, for all aAR; we have lrra ¼ alr: For all mAM; the series lrðmÞ converges, and yields an R module homomorphism from Mr to L:

Proof. We first observe that the commutator of p in Lfftgg is just L: This is straightforward. Suppose now that we have constructed a series l such that DðlÞ¼1 and lrp ¼ pl: We claim lra ¼ al for all aAR: To see this, choose an aAR and observe that lrarp ¼ 0 0 À1 0 plra: Let l ¼ lra: We deduce that l l commutes with p: Thus, lra ¼ l ¼ cl with cAL: Comparing constant terms shows that c ¼ a; which proves that lra ¼ al: We find l by the method of undetermined coefficients. Set XN XN n n rp ¼ p þ bnt and l ¼ 1 þ cnt : n¼1 n¼1

From the relation lrp ¼ pl; we deduce the following recurrence relations for determining the coefﬁcients cn: X qn q j ðp À p Þcn ¼ cjbi : i þ j ¼ n jon

qn Since p ap for all nX1; the coefﬁcients cn (and hence l) are uniquely determined by these relations. P qn It remains to show that the series lðxÞ¼ n cnx converges on all of M: Let w be the normalized additive valuation on L: Then e ¼ wðpÞ is the ramiﬁcation index of L=K: We claim that wðcnÞX À ne for all nX0: This is clearly true for n ¼ 0: It follows easily by induction from the above recurrence relations for the cn: Assume mAM: Then wðmÞ40; and we see

qn n n n Àn wðcnm Þ¼wðcnÞþq wðmÞX À ne þ q wðmÞ¼q ðwðmÞÀenq Þ:

As n-N; this expression tends to N; which proves that the series lðmÞ converges. &

We remark that lr need not be one-to-one. Indeed, if mAMr is a torsion element, n we have rpn ðmÞ¼0 for some n40: Applying lr to both sides, we ﬁnd that p lrðmÞ¼ 0; which implies lrðmÞ¼0: We will soon show that the kernel of lr; as a homomorphism on Mr; is exactly the submodule of torsion elements. Next, we will construct an inverse to lr that will be like an exponential function. À1A Let er ¼ lr Lfftgg: From the key property lrra ¼ alr; we deduce the key property of er; era ¼ raer: ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 241

Proposition 2.2. Let rADˆ RðOÞ: There is a unique element erALfftgg such that DðerÞ¼1 and era ¼ raer for all aAR: Let e ¼ eðL=KÞ; the ramiﬁcation index of L=K: # Then erðxÞ converges on M ¼fmAM j wðmÞ4e=ðq À 1Þg: It deﬁnes an R module # homomorphism from M to Mr:

À1 Proof. We have seen that there is an element,P namely er ¼ lr ; such that DðerÞ¼1 A n and era ¼ raer for all a A: Write er ¼ 1 þ nX1 dnt : We will derive a recursion relation for the coefﬁcients dn; which will establish the uniqueness. Also, it will enable us to determine the region of convergence. From erp ¼ rper; we deduce the relation X qn qi ðp À pÞdn ¼ bidj : i þ j ¼ n jon

Since d0 ¼ 1 by hypothesis, this shows the coefﬁcients dn (and thus er) are uniquely determined. nÀ1 We claim that wðdnÞX Àð1 þ q þ ? þ q Þe for nX1: We prove this by q induction. For n ¼ 1; we have ðp À pÞd1 ¼ b1; and so wðd1ÞX À e: For nX2; we ﬁnd

qn i e þ wðdnÞ¼wððp À pÞdnÞX min ðq wðdjÞÞ: i þ j ¼ n jon

By induction, we ﬁnd

i i nÀiÀ1 i iþ1 nÀ1 q wðdjÞX À q ð1 þ q þ ? þ q Þe ¼Àðq þ q þ ? þ q Þe

X Àðq þ q2 þ ? þ qnÀ1Þe:

Putting the last two equations together proves the estimate. Now, suppose wðmÞ4e=ðq À 1Þ: Then n Àn n q À 1 1 À q wðd mq ÞX À e þ qnwðmÞ¼qn wðmÞÀ e : n q À 1 q À 1

This expression tends to inﬁnity as n-N: Thus, erðmÞ converges. qn The remaining assertion is clear, since the above estimate shows that dnm AM for all nX0: &

If r40 is a positive integer, it is easy to see that for any aAR; we have r D r r raðM Þ M : It follows that ðM Þr is an R submodule of Mr: Another useful remark r rþ1 is that rpðM ÞDM : ARTICLE IN PRESS

242 M. Rosen / Journal of Number Theory 103 (2003) 234–256

Proposition 2.3. Let rADˆ RðOÞ: Suppose that r is an integer, and that r is greater than e=ðq À 1Þ (here e ¼ eðL=KÞ). Then lr and er are inverse R module isomorphisms of r r r ðM Þr with M : In particular, ðM Þr is a torsion-free R module.

r Proof. If mAM with r4eðL=KÞ=ðq À 1Þ; we will show that wðerðmÞÞ ¼ wðmÞ and wðlrðmÞÞ ¼ wðmÞ: This will sufﬁce to prove the proposition, because lr and er are formally inverse to one another.P qn Recall that erðmÞ¼m þ nX1 dnm : Our claim about er will follow if we can show

qn wðmÞowðdnm Þ8n40:

By the result proved in the course of the proof of Proposition 2.2, we have

n qn n n q À 1 wðdnm Þ¼wðdnÞþq wðmÞXq wðmÞÀ e: q À 1

Since wðmÞ4e=ðq À 1Þ; this last quantity is 4qnwðmÞÀðqn À 1ÞwðmÞ¼wðmÞ: This establishes the proposition as it applies to er: The proof for lr is similar and even simpler, so we omit it. &

Corollary. If eðL=KÞoq À 1; then Mr is isomorphic to M; and so Mr is torsion-free.

Proof. In this case, both the exponential and the logarithm for r converge on all of M: &

Proposition 2.4. Let rADˆ RðOÞ: Let mAM: Then lrðmÞ¼0 if and only if m is a torsion element.

Proof. We have already proven the ‘‘if’’ part of the assertion. Suppose now that lrðmÞ¼0: Let r be an integer greater than e=ðq À 1Þ: By a r A rþ1C r previous remark, we know that rpr ðmÞ¼rpðmÞ M M : Then lrðrpr ðmÞÞ ¼ r r p lrðmÞ¼0: But lr is an isomorphism on M : Thus, rpr ðmÞ¼0; i.e. m is a torsion element. &

Theorem 2.5 (Local uniform boundedness theorem). Let L range over all ﬁnite extensions of K of degree N: Let OL be the integral closure of R in L; and let ML be the maximal ideal of OL: Choose such an L; and let r be any formal Drinfeld R module deﬁned over OL: Then the size of the R torsion subgroup of ðMLÞr is bounded by qN=ðqÀ1Þ:

Proof. Let T be the R torsion subgroup of ðMLÞr; and let r ¼½eðL=KÞ=ðq À 1Þ þ 1: r r By Proposition 2.3, ðMLÞr is R torsion-free. Thus, T injects into ML=ML: The size of this latter group is qðrÀ1Þf ðL=KÞ; where f ðL=KÞ is the relative degree. The exponent is ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 243 estimated by

ðr À 1Þf ðL=KÞ¼½eðL=KÞ=ðq À 1Þ f ðL=KÞpeðL=KÞf ðL=KÞ=ðq À 1Þ

¼ N=ðq À 1Þ: &

The ultimate reason for this uniformity is that the exponential function er converges on a domain that depends on the ramification, but not on the formal Drinfeld module r: We note that in characteristic p; if pjN there are infinitely many extensions L of K of degree N: By the use of local class field theory, we may even choose these extensions to be abelian over K: For p-adicfields, there are only finitely many extensions of degree N in a fixed algebraicclosure.

3. In this section, we prove a Weierstrass preparation theorem for the non- commutative local ring O ¼ Offtgg: This is a special case of a result of Honda on twisted power series rings, see [10] and/or Section 20.3 of [9]. Honda’s proof is by successive approximations. We present an alternative proof, which is inspired by a method of Manin in the case of commutative power series rings. The maximal ideal M of O gives rise to a two-sided ideal MO of O: MO is the kernel of the natural homomorphism O-Efftgg: We give O the topology deﬁned by making the powers of MO into a neighborhood basis of 0: As is easily seen, O is complete in this topology. Fix a positive integer N; and deﬁne a mapping t from O to O as follows: ! XN XN n nÀN t bnt ¼ bnt : n¼0 n¼N

We note that t has the following properties: t is O linear; for all hAO; tðhtN Þ¼h; tðhÞ¼0 if and only if degtðhÞoN: Both in the statement of the following theorem, and throughout the proof, we will pay careful attention to the order of the factors. This is necessary, since O is non- commutative. P N nA A A Theorem 3.1. Let f ¼ n¼0 bnt O be such that bn M for noN and bn O is a unit. Given any gAO; there exist unique elements q; rAO such that g ¼ qf þ r; with either r ¼ 0 or degtðrÞoN:

Proof. Let P be a generator of the maximal ideal M: By hypothesis, we can write f in the form f ¼ PP þ utN ; where PAOftg is a polynomial in t of degree less than N and u is a unit in O: We could at this point simply present a formula for q; and verify that it has the required property. However, for the sake of motivation, we prefer to work backwards. Assume we already have q and r: The relation g ¼ qf þ r implies tðgÞ¼ tðqf Þ: Using the above expression for f ; we ﬁnd qf ¼ qPP þ qutN : Thus, tðgÞ¼ ARTICLE IN PRESS

244 M. Rosen / Journal of Number Theory 103 (2003) 234–256 tðqPPÞþqu; and so tðgÞuÀ1 ¼ tðqPPÞuÀ1 þ q: Let us define an O-linear operator E on O by the formula EðhÞ¼tðhPPÞuÀ1: Then XN tðgÞuÀ1 ¼ðI þ EÞðqÞ so that q ¼ ðÀ1ÞiEiðtðgÞuÀ1Þ: i¼0 We now turn matters around. Let this formula be the definition of q: First we show that the infinite series converges, and then we show that the sum, q; has the desired property. The convergence is quite easy. For any hAO; we readily see there is an hÃO such that hP ¼ Ph˜: Thus, EðhÞ¼tðhPPÞuÀ1 ¼ tðPhP˜ ÞuÀ1 ¼ PtðhP˜ ÞuÀ1AMO: By induction, EiðhÞAMiO: It follows that the series for q converges. By the calculation of the last paragraph, if hAMiO; then EðhÞAMiþ1O: It follows that E is a continuous linear operator on O: Thus, ! XN XN EðqÞ¼E ðÀ1ÞiEiðtðgÞuÀ1Þ ¼ ðÀ1ÞiEiþ1ðtðgÞuÀ1Þ: i¼0 i¼0

If we now add q and EðqÞ; we ﬁnd that q þ EðqÞ¼tðgÞuÀ1: Equivalently, qu þ tðqPPÞ¼tðgÞ: Since f ¼ PP þ utN ; qf ¼ qPP þ qutN ; and tðqf Þ¼tðqPPÞþqu: This shows that tðqf Þ¼tðgÞ: By the third property of the operator t; we have g ¼ qf þ r; where either r ¼ 0 or degtðrÞoN: It remains to show the uniqueness of q and r: Assume that there exist two other elements q0; r0AO with the same properties as q and r: Then ðq0 À qÞf ¼ r À r0: If q0aq; we can write q0 À q ¼ Pmv; where v has non-zero reduction modulo M: It follows that r À r0 ¼ Pmw; where w is either 0 or a non-zero element of Oftg of degree less than N: Dividing both sides by Pm and reducing modulo M; we ﬁnd v%f%¼ w%: Since the left-hand side is a non-zero power series in t divisible by tN ; and the right-hand side is either zero or a polynomial of degree less than N; this is a contradiction. It follows that q ¼ q0 and r ¼ r0: &

Definition. A polynomial QAOftg such that degtðQÞ¼N is called a distinguished polynomial if it is monic and all its coefficients are in M except for the leading coefficient. P N nA Theorem 3.2 (The Weierstrass Preparation Theorem). Let f ¼ n¼0 bnt O be such that bnAM for noN and bN AO : Then there is a unit U and a distinguished polynomial Q in O such that f ¼ UQ: We have degtðQÞ¼N: The unit U and the distinguished polynomial Q are uniquely determined.

Proof. Apply Theorem 3.1 to f and the polynomial g ¼ tN : We ﬁnd tN ¼ qf þ r; % where r is either 0 or degtðrÞoN: Reducing modulo M shows that r% ¼ 0; i.e. all the coefﬁcients of r are in M: Thus, Q ¼ tN À r is a distinguished polynomial of degree N: From Q ¼ qf; we see, again by reducing modulo M; that the constant ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 245 term of q is a unit in O: Thus, q is a unit in O: Setting U ¼ qÀ1; we have f ¼ UQ; as asserted. The uniqueness follows from the uniqueness assertion of Theorem 3.1. We omit the details. &

Let rADˆ RðOÞ be a formal Drinfeld module. We want to apply the above theorem to the determination of the structure of the torsion points associated with r: Let L0 be a ﬁnite algebraicextension of L; O0 the integral closure of O in L0; and M0 the maximal ideal of the local ring O0: We can make M0 into an R module via r: 0 0 We denote M with this new action as an R module by M r: This procedure also works for any algebraicextension L0=L; because, for any m0AM0; we have m0ALðm0Þ; 0 0 which is complete. Thus, for all aAR; raðm Þ converges to an element of M : Let Msep be the maximal ideal in the ring of integers of the maximal separable extension, Lsep; of L:

Theorem 3.3. Let rADˆ RðOÞ be a formal Drinfeld module with stable reduction of sep n height h40: Let p be a generator of P; the maximal ideal of R: Denote by Mr ½p the sep n submodule of Mr annihilated by p : Then,

sep n n" n"?" n Mr ½p is isomorphic to R=P R=P R=P ðh timesÞ as R modules.

Proof. By the deﬁnition of the height, we have ordtðr% pÞ¼h deg P: It follows that ordtðr% pn Þ¼nh deg P: We apply the Weierstrass Preparation Theorem to the series rpn : We ﬁnd a unit U and a distinguished polynomial Q of degree N ¼ nh deg P such that rpn ¼ UQ: Note that this implies that the constant term of Q is a generator of Pn: sep The roots of rpn ðxÞ on M coincide with the roots of QðxÞ¼0: Write Q ¼ 2 N b0 þ b1t þ b2t þ ? þ t : Then,

q q2 qN QðxÞ¼b0x þ b1x þ b2x þ ? þ x :

Since the derivative with respect to x of QðxÞ is b0a0; we see that QðxÞ is a separable polynomial. Moreover, since biAM for 0pioN; we see that all the roots of QðxÞ in Lsep are automatically in Msep: It follows that the number of elements in sep n N nh deg P nh Mr ½p is q ¼ q ¼ NP ; where NP ¼ #ðR=PÞ: sep Consider the case where n ¼ 1: Since Mr ½p is a ﬁnite R module annihilated by p; it is a sum of a ﬁnite number of copies of R=P: Since the number of elements in this submodule is NPh; we see it is isomorphicto the sum of h copies of R=P: The result for n41 follows from what has been shown so far combined with the structure theory of torsion modules over a principal ideal domain. & ARTICLE IN PRESS

246 M. Rosen / Journal of Number Theory 103 (2003) 234–256

We conclude this section by raising two questions concerning the Galois module structure of the torsion modules discussed in the above theorem. sep n To ease the notation, let us fix r; and define Ln ¼ Mr ½p : Since rp has sep coefficients in L; Ln is a module for the action of G :¼ GalðL =LÞ: Since Ln consists of all the roots of a separable polynomial, it follows that LðLnÞ is a Galois extension of L: Putting this together with the result of Theorem 3.3, we find a monomorphism

n GalðLðLnÞ=LÞ-AutRðLnÞDGlhðR=P Þ: S We can also pass to the limit. Let LN ¼ n Ln: We derive a monomorphism

GalðLðLNÞ=LÞ-GlhðRÞ:

It is, of course, of interest to investigate the images of these maps. The associated Tate module, TRðrÞ; is deﬁned as the inverse limit of the modules sep n Mr ½p with respect to the maps ‘‘multiplication by p’’. It is a free module of rank h over R; and a module for G; the Galois group of Lsep=L: In [1], the author proves a local version of the Tate isogeny conjecture for these modules. In fact, he proves this in the more general context of t-modules. Now, suppose h ¼ 1: Then we have a monomorphism

GalðLðLNÞ=LÞ-Gl1ðRÞ¼R :

Thus, when the height is 1; LðLNÞ is an abelian extension of L: Let us specialize further. Assume O ¼ R and deg P ¼ 1; i.e. E ¼ F ¼ Fo: For example, we can choose ˆ rp ¼ p þ t; and extend this to an element of D RðRÞ: One can then prove that GalðKðLNÞ=KÞDR : It is even true that the maximal abelian extension of K can be obtained as the compositum of KðLNÞ and the maximal unramiﬁed extension of K: At this point, the proof of this assertion uses the theory of Lubin–Tate formal groups. It is a challenge to ﬁnd a proof which stays entirely within the context of formal Drinfeld modules.

4. Up to now we have been working exclusively in a local setting. In this section, we will start with a Drinfeld module r and a place P of a global function field. We construct, subject to a mild restriction, a formal Drinfeld module r#; the formal completion of r at P: We will show how this is useful for understanding torsion points on r: The whole process is patterned on the construction of a formal group associated to an elliptic curve at a finite place of a global field. Let k be a global function field whose field of constants, Fo; has q elements. As usual, p will denote the characteristic of k: Let N be a place of k; and let ACk be the subring consisting of elements whose only pole is at N: If E=k is a field extension of k; we denote by DAðEÞ the set of Drinfeld A modules defined over E: To recall, rADAðEÞ if it is a homomorphism from A to Eftg such that DðraÞ¼a for all aAA and rðAÞD/ E: We assume that the reader is familiar with the elementary theory of Drinfeld modules (see [7,8] or [14]). We briefly review some definitions. ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 247

Each Drinfeld module has associated to it a positive integer r called its rank. It is characterized by the condition degtðraÞ¼r degðaÞ for all aAA: The degree of a; degðaÞ; is defined by the relation qdegðaÞ ¼ #ðA=aAÞ: Let B be the integral closure of A in E; and M a maximal ideal of B: Suppose - raABM ftg for all aAA: Reduction modulo M makes a r% a a Drinfeld A module over B=M provided that r%ðAÞD/ B=M: Under these conditions, we say that r has stable reduction at M: More generally, if rADAðEÞ; we say that r has stable reduction at M if it is isomorphicover E to a Drinfeld module with these two properties. If r has stable reduction at M; and the rank of r% is equal to the rank of r; we say r has good reduction at M: Since A is a finitely generated Fo algebra, it is easily seen that r has good reduction at all but finitely many primes of B: In fact, something stronger is true. For almost all primes M; we have raABM ftg for all aAA; and the leading coefficient of ra is a unit in BM : Assume that r has stable reduction at M: We may suppose that raABM ftg for all aAA: Then there is a positive integer h (depending on M) such that ordtðr% aÞ¼ h ordPðaÞdeg P for all aAA: Here, P ¼ M-A: The integer h is called the height of r at M: We can now proceed with the definition of the formal completion of r: Let us continue to assume that r has stable reduction at M and that raABM ftg for all aAA: À1 If sAA and seP; then rs is a unit in BM fftgg: Denote its inverse by rs : If rAAP; À1 write r ¼ a=s where a; sAA and seP: Map r to rars : It is easily checked that this is a well defined map, and is a homomorphism from AP to BM fftgg: Let us continue to use the letter r for this extended map. Notice that DðrrÞ¼r for all rAAP: ˆ Let Bˆ M be the completion of B in the M-adictopology, and A P be the completion of A in the P-adictopology. From the embedding of BM CBˆ M ; we get a homomorphism from AP to Bˆ M fftgg: As we have seen earlier, the latter ring is a local ring which is complete in the topology induced by the powers of the maximal ideal. Since our homomorphism is easily seen to be continuous, r extends to a # ˆ - ˆ homomorphism r : A P B M fftgg; which has the property that DðrrÞ¼r for all ˆ rAA P: Also note that r# coincides with r on A: It follows that r# is a formal Drinfeld ˆ A P module defined over Bˆ M :

Deﬁnition. Let rADAðEÞ have stable reduction at a prime MCB; the integral ˆ closure of A in E: Assume r has coefﬁcients in BM : The formal Drinfeld A P module constructed above is called the formal completion of r at M:

Suppose r has stable reduction at M; but that its coefficients are not in BM : By definition, it is isomorphicover E to a Drinfeld module whose coefficients are in BM and for which we can carry out the above construction. If we do this with two different isomorphisms, one can show that the resulting formal Drinfeld modules are isomorphic. We omit the proof. ˆ To bring the notation into line with that of earlier sections, let us set A P ¼ R; ˆ K ¼ k P; Bˆ M ¼ O; and Eˆ M ¼ L: We let Pˆ be the maximal ideal of R and Mˆ be the ARTICLE IN PRESS

248 M. Rosen / Journal of Number Theory 103 (2003) 234–256 maximal ideal of O: This is slightly different than our previous notation, but it should cause no confusion. Finally, we set E ¼ O=Mˆ : Assuming that r has coefﬁcients in BM and stable reduction, we have the following exact sequence of A modules:

ð0Þ-Mˆ r# -Or-Er% -ð0Þ: ð1Þ

The action of A on Mˆ via r has been extended to the action of R on Mˆ via r#: Thus,

Mˆ r# denotes Mˆ with the action of R given by the formal Drinfeld module r#: This shows that Mˆ r# considered as an A module has no Q torsion for any prime ideal Q # not equal to P: In fact, if aAA but aeP; then ra ¼ ra is a unit in Offtgg: This has an immediate global consequence.

Proposition 4.1. If rADAðEÞ; then Er½tors is ﬁnite.

Proof. Let M1 and M2 be two primes of B for which r has good reduction. Suppose A a r DAðBM1 Þ-DAðBM2 Þ; and P ¼ M1-A M2-A ¼ Q: Any element in Er½tors is integral over BM1 and over BM2 ; and so must be in both rings. By the remark preceding the proposition, we see that the kernel of the obvious - " - " homomorphism from Er½tors ðBM1 Þr ðBM2 Þr ðB=M1Þr% 1 ðB=M2Þr% 2 is trivial. Thus, #ðEr½torsÞpNM1 Â NM2: &

Remark. We give a second proof of the same result. Let M be a prime of B such that rADAðBM Þ and such that r has good reduction at M: Then, as before, we see that C C Er½tors ðBM Þr Or: It sufﬁces to show that Or½tors is ﬁnite. From the exact sequence ð1Þ we derive

ð0Þ-Mˆ r# ½tors-Or½tors-Er% :

½L:K=ðqÀ1Þ By Theorem 2.5, the order of Mˆ r# ½tors is bounded by q : It follows that

½L:K=ðqÀ1Þ #Or½torspNM Â q : ð2Þ

Of course, ½L : Kp½E : k: This yields a bound on the size of Er½tors which depends only on the degree of E over k and the norm of one prime of good reduction. We will come back to this in the next section. Let Lsep be the maximal separable extension of L in a fixed algebraicclosure, Osep the integral closure of O in Lsep; and Mˆ sep the maximal ideal of Osep: Finally, let Ealg be the algebraicclosure of E ¼ O=Mˆ : The following exact sequence of A modules is derived from Eq. (1) by applying it first to finite separable extensions of L and then passing to the direct limit:

- ˆ sep- sep- alg- ð0Þ M r# Or Er% ð0Þ: ð3Þ ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 249

Proposition 4.2. Let rADAðBM Þ; and suppose that r has good reduction at M: Let X ˆ sep n % n 1 be an integer. As an A module, M r# ½P is isomorphic to the direct sum of h copies of A=Pn; where h% is the height of r%: Moreover, the following sequence is exact:

- ˆ sep n - sep n - alg n - ð0Þ M r# ½P Lr ½P Er% ½P ð0Þ:

Proof. We will prove both assertions simultaneously. sep Since r is assumed to have good reduction at M; every torsion point of Lr is sep n sep n integral over BM ; and is thus integral over O: It follows that Lr ½P ¼Or ½P : From Eq. (3), we derive that

- ˆ sep n - sep n - alg n ð0Þ M r# ½P Lr ½P Er% ½P is exact. sep n n By the theory of Drinfeld modules, Lr ½P is isomorphicto r copies of A=P ; alg n % n where r is the rank of r: Also, Er% ½P is isomorphicto r% À h copies of A=P ; where r% is the rank of r% and h% is its height. Since r has good reduction, we must have r ¼ r%: ˆ sep n % To prove the proposition it sufﬁces to show that M r# ½P is the direct sum of h copies of A=Pn: Let h be the height of the reduction of r# modulo Mˆ : We ﬁrst show that h ¼ h%: Let 2 ˆ pAP À P : Then p is a uniformizing parameter for R ¼ A P: Also, since pAA; we # # ˆ have rp ¼ rp: It follows that the reduction of rp at M coincides with the reduction of % rp at M; namely r% p: Thus, h is given by ordtðr% pÞ=deg P: This is also h; because % % ordtðr% pÞ¼h ordPðpÞdegðPÞ¼h degðPÞ: ˆ sep n ˆ nD n By Theorem 3.3, M r# ½p is the sum of h copies of R=P A=P (note that what we called P in Theorem 3.3 is here denoted Pˆ ). Since h ¼ h%; we will be done if we can ˆ sep n ˆ sep n ˆ sep show M r# ½p ¼M r# ½P : This is easy. The A action on M r# extends to the action of R; as we have seen. Thus the Pn torsion is the same as the Pˆ n torsion. Since Pˆ n is generated by pn; the proof is complete. &

5. We maintain the notations of Section 4. The uniform boundedness conjecture asserts that for all Drinfeld modules r of fixed rank r in DAðEÞ; there is a constant C40 such that #Er½torspC: This was first enunciated in the context of Drinfeld modules by Poonen [13]. The motivation, of course, is the corresponding result for ellipticcurves defined over Q: Poonen proved (in several different ways) this conjecture when r ¼ 1: As far as I know, the corresponding result for rank two or greater is still open. See [15] for some recent contributions. In this section, we will prove a general result depending on a hypothesis concerning potentially good reduction. A corollary will be a somewhat different proof of Poonen’s theorem about Drinfeld modules of rank one. A second, more specialized, result will concern Drinfeld modules of rank two. This will allow bad reduction which is not ‘‘too bad’’. ARTICLE IN PRESS

250 M. Rosen / Journal of Number Theory 103 (2003) 234–256

ðrÞ Let S be a finite set of primes of A: Define DA ðE; SÞ to be the set of Drinfeld A modules of rank r; defined over E; which have potentially good reduction at some prime of B lying above a prime in S:

Theorem 5.1. There is a constant C; depending on r; E; and S; with the property that, A ðrÞ for all r DA ðE; SÞ; we have #Er½torspC:

Proof. One easily reduces to the case where S consists of one prime P: Let A ðrÞ C r DA ðE; fPgÞ; and let M B be a prime lying above P at which r has potentially good reduction. It follows from the theory of reduction at a prime (see [3] or [16]) that there is an integer n; depending only on the rank r; such that if E0 is a field extension of E totally ramified of degree qn À 1 at all primes M0 over M; then r has good reduction at every such M0 over E0: Since we can use the same field E0 for every Drinfeld module under consideration, we might as well assume E0 ¼ E; and that r already has good reduction at M: In this case, we can use the remark following Proposition 4.1 to conclude that

½E:k=ðqÀ1Þ #Er½torspNM Â q : &

A ð1Þ Corollary. There is a constant C such that, for all r DA ðEÞ; we have #Er½torspC:

Proof. By a result of Drinfeld (see [3] and/or [16]), we know that every Drinfeld module of rank one has potentially good reduction at any prime of its ﬁeld of deﬁnition. One can thus choose any prime PCA; set S ¼fPg; and apply the theorem. &

We are now going to consider Drinfeld modules of rank two. At a given prime, such modules either have potentially good reduction or potentially stable reduction. In the latter case, there is an explicit local parametrization, the Drinfeld–Tate parametrization. This allows one to give a measure of how bad the bad reduction is. From now on we assume A ¼ F½T; the ring of polynomials in one variable. Every 2 element r in DAðEÞ of rank two is completely determined by rT ¼ T þ ct þ Dt ; qþ1 where c; DAE and Da0: Set jr ¼ c =D: This is the j-invariant of r: It plays the same role in the theory of Drinfeld modules of rank two as the j-invariant in the theory of ellipticcurves.Two Drinfeld modules of rank two have the same j- invariant if and only if they are isomorphicover some finite algebraicextension of their field of definition. A detailed exposition can be found in [5].

Lemma 5.2. Let rADAðEÞ be a rank two Drinfeld module. Let M be a prime ideal of the integral closure B of A in E: Then r has potentially good reduction at M if and only if ordM ð jrÞX0:

Proof. By a result of Drinfeld (see [16]), r has potentially stable reduction at M: This means that there is a ﬁnite extension E0 of E and a prime M0 of E0 lying over M with ARTICLE IN PRESS

M. Rosen / Journal of Number Theory 103 (2003) 234–256 251 the following properties. There is a Drinfeld module r0; isomorphicto r over E0; such that the coefficients of r0 are integral at M0; and the reduction of r0 modulo M0 is a 0 Drinfeld module (possibly of lower rank) over the residue class field. Let r T ¼ 0 0 2 0 0 0 0 0 T þ c t þ D t : Clearly, not both c and D can lie in M : Either ordM0 ðD Þ¼0 and r 0 0 has good reduction, or ordM0 ðD Þ40 and ordM0 ðc Þ¼0: In the first case ordM0 ð jr0 ÞX0; and in the second case, ordM0 ð jr0 Þo0: Since jr ¼ jr0 ; we see that ordM ð jrÞX0 if and only if r has potentially good reduction, as asserted. &

For the moment, ﬁx c and D: Suppose that c; DABM ; ordM ðDÞ40; and ordM ðcÞ¼ 2 0: Let rADAðEÞ be determined by rT ¼ T þ ct þ Dt : Notice that with these conditions ordM ð jrÞ¼ÀordM ðDÞ: We pass to the completion, O ¼ Bˆ M : Drinfeld has shown how to giveP a local parametrization of r as follows (see [3] or [5]). There is N nA A a power series e ¼ n¼0 unt Offtgg with u0 ¼ 1; and a Drinfeld module f DAðOÞ % of rank one, such that for all aAA; era ¼ fae: Moreover, f ¼ r%: One sees that f has good reduction. Finally, set L equal to the quotient ﬁeld of O: Then

q q2 eðxÞ¼x þ u1x þ u2x þ ? ð1Þ deﬁnes a homomorphism, which is an entire function, from Lalg-Lalg: The kernel, G; of eðxÞ is a one-dimensional f-lattice. This means that: G is a discrete additive alg subgroup of L ; for all aAA; we have faðGÞDG; and that there is a non-torsion element gAG such that fðAÞg ¼ G: Finally, G is mapped into itself by GalðLalg=LÞ: The Drinfeld module r is said to be parametrized by the pair ðf; GÞ: We will call g the M-adicperiod of r: It is uniquely determined up to multiplication by an element of qÀ1 F : It is easily seen that g AL: One must have ordM ðgÞo0: Otherwise, G could not be discrete.

Lemma 5.3. Let r ¼ rc;D be the rank two Drinfeld module over E determined by 2 rT ¼ T þ ct þ Dt : Suppose: M is a prime of B; ordM ðcÞ¼0; and ordM ðDÞ40: Let g denote its M-adic period as deﬁned above. Then

ordM ð jrÞ¼ÀordM ðDÞ¼ðq À 1ÞordM ðgÞ:

Proof. Let ðf; GÞ be the Drinfeld–Tate parametrization of r as above. By the analytic theory of complete valued ﬁelds, we can write the following product expansion for eðxÞ: Y x eðxÞ¼x 1 À : ð2Þ f ðgÞ aAA a aa0

Let w denote the unique extension of the additive valuation ordM from E to L and alg thence to L : Since faðxÞ has coefﬁcients in O; and since wðgÞo0; we deduce that ARTICLE IN PRESS

252 M. Rosen / Journal of Number Theory 103 (2003) 234–256

degðaÞ wðfaðgÞÞ ¼ q wðgÞ: Comparing the expansion of eðxÞ as a sum as in Eq. (1) and n as a product as in Eq. (2), we deduce that wðunÞX Àðq À 1ÞwðgÞ: We will use this inequality a little further on. Note that wðunÞ40 for all nX1: From the relation eðfaðxÞÞ ¼ raðeðxÞÞ; we deduce that eðxÞ gives an isomorphism À1 alg between fa ðGÞ=G and the a-torsion subgroup of Kr : It follows that Y u r ðuÞ¼au 1 À : a eðg0Þ 0 À1 g Afa ðGÞ=G g0a0

In particular, setting a ¼ T; we ﬁnd the following expression for D; T D ¼ 7 Q ; ð3Þ eðg0Þ

0 À1 where g runs over the non-zero elements of rT ðGÞ=G: We can ﬁnd a nice set of representatives for this set. q q Let fT ðuÞ¼Tu þ bu ; where b in a unit in O; and let b be a solution to Tu þ bu ¼ À1 g: Then the following set is a set of representatives for fT ðGÞ=G;

fab þ l j aAF; l such that fT ðlÞ¼0g:

2 It sufﬁces to notice that fT ðab þ lÞ¼ag þ 0AG; that this set has exactly q elements, and that no two elements of this set are congruent modulo G: The latter fact is an easy consequence of wðlÞX0 and wðbÞo0 (see the next paragraph). We note that wðbÞo0; since otherwise wðgÞX0; which is false. Also, wðlÞX0; since q l is the root of fT ðxÞ¼Tx þ bx ; where b is a unit. We will show later that wðeðbÞÞ ¼ wðbÞo0; and that wðeðlÞÞ ¼ wðlÞX0: Let us assume these assertions for now. Then eðab þ lÞ¼aeðbÞþeðlÞ: It follows that if aa0; wðeðab þ lÞÞ ¼ wðbÞ: If a ¼ 0 and la0; then wðeðab þ lÞÞ ¼ wðlÞ: From Eq. (3) we now deduce X wðDÞ¼wðTÞÀðq2 À qÞwðbÞÀ wðlÞ: l

The sum is over all non-zero roots of Tu þ buq ¼ 0: The product of these non-zero roots is bÀ1T; so the sum reduces to wðTÞ: It follows that wðDÞ¼Àðq À 1ÞqwðbÞ: Since bbq þ Tb ¼ g; we see that qwðbÞ¼wðgÞ: Thus we have wðDÞ¼Àðq À 1ÞwðgÞ; as asserted. It remains to prove that wðeðlÞÞ ¼ wðlÞ and wðeðbÞÞ ¼ wðbÞ: We prove these in order. We haveP seen that wðlÞX0: Since we have shown earlier that all the coefﬁcients of qn eðxÞ¼ n unx ; with nX1; are in the maximal ideal of O; it follows that qn wðlÞowðunl Þ for all nX1: Thus, wðeðlÞÞ ¼ wðlÞ: n We have already shown that qwðbÞ¼wðgÞ; and that wðunÞX Àðq À 1ÞwðgÞ for all qn n n nÀ1 nX0: This implies that wðunb Þ¼wðunÞþq wðbÞX Àðq À 1ÞwðgÞþq wðgÞ¼ ARTICLE IN PRESS

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ðqnÀ1 À qn þ 1ÞwðgÞ: For nX1; this quantity is non-negative. On the other hand, wðbÞo0: Thus, wðeðbÞÞ ¼ wðbÞ; which is what we set out to prove. &

We are now in a position to prove the following generalization of Theorem 5.1 in the case of rank two Drinfeld modules over a polynomial ring.

Theorem 5.4. Let A ¼ F½T; k ¼ FðTÞ; and let E be a finite algebraic extension of k: Let S be a finite set of primes of A: For N40; we denote by DAðE; S; NÞ the set of rank two Drinfeld A modules, r; which are defined over E; having the property that for some prime M of E; lying over a prime in S; we have ordM ð jrÞX À N: There is a constant CN (depending on E and S as well as N) such that #Er½torspCN for all rADAðE; S; NÞ:

Proof. We note that by Lemma 5.2, the case N ¼ 0 is a special case of Theorem 5.1. The integer N is a measure of the ‘‘badness’’ of the bad reduction allowed for r at primes above those in S: We begin the proof by making several reduction steps. To begin with, it is clear that it suffices to prove the result when S consists of one prime fPg: We assume this from now on. Secondly, let E1 be a finite algebraicextension 0 of E such that if M is a prime of E1 lying over P; then the ramification index eðM0=MÞ is divisible by q2 À 1 (here M ¼ M0-E). The existence of such a field is easily seen. From [3] (or [16]), one sees that every rADAðEÞ of rank two has stable reduction at every prime of E1 lying above P: Since ½E1 : EoN; it does no harm to assume E ¼ E1: By the discussion in the last paragraph, we can assume for rADAðE; fPg; NÞ that 2 there is a prime M of E lying over P; such that rT ¼ T þ ct þ Dt with c; DABM and either ordM ðDÞ¼0 or ordM ðDÞ40 and ordM ðcÞ¼0: In the former case, r has good reduction at M; and there is a bound on Er½tors by Theorem 5.1. So, we can assume we are in the case where ordM ðDÞ40 and ordM ðcÞ¼0: In this case, we have a Drinfeld–Tate parametrization ðf; GÞ; where f is a rank one Drinfeld A module defined over O ¼ Bˆ M having good reduction at Mˆ ; and G ¼ fðAÞg is a rank one f- alg lattice in L : Here, L ¼ Eˆ M is the quotient field of O: Under our assumptions, ordM ð jrÞ¼ÀordM ðDÞX À N: It follows that 04ðq À 1ÞordM ðgÞX À N (see Lemma 5.3). À1 As in the proof of Lemma 5.3, we see that eðxÞ gives an isomorphism of fa ðGÞ=G alg a A alg with Lr ½a; for any 0 a A: There is a nice set of representatives in Lr for the À1 quotient group, fa G=G: Let la be a primitive fa division point, and let ga be a root of faðxÞ¼g: Then

ffcðlaÞþfd ðgaÞjdegðcÞ; degðdÞodegðaÞg is such a set of representatives. We claim fa applied to any element in this set is an element of G; and that the set contains q2degðaÞ elements. Also, no two elements of this set are congruent modulo G: These assertions are easily veriﬁed. Note that ARTICLE IN PRESS

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a wðfcðlaÞÞX0; and, if d 0; wðfd ðgaÞÞo0: As in the proof of Lemma 5.3, w denotes the unique extension of ordM ðxÞ to the algebraicclosure of L: The proof will proceed as follows. First we will consider the case where a ¼ c; a prime polynomial. We will show that there are only ﬁnitely many prime polynomials such that Lr½cað0Þ: The set of such primes will depend on N; E; and S; but not on the Drinfeld modules r satisfying the hypotheses of the theorem. Fixing a prime polynomial c; we will consider the powers a ¼ cm of c; and show that there is an upper bound on the values of m (again depending only on N; E; and S) such that Lr m contains a primitive c -division point. Since ErCLr; these two results will establish the theorem. Now, let us assume that a ¼ c; an irreducible polynomial, and that

a 0 x ¼ eðfcðlcÞþfd ðgcÞÞ is an c-division point for r lying in L: If d is not divisible by c; there is a d0 such that 0 c dd 1 ðmod Þ: It follows that rd0 ðxÞ¼eðfd0cðlaÞÞ þ eðgcÞ; and so wðrd0 ðxÞÞ ¼ wðeðgcÞÞ ¼ wðgcÞ: The last equality follows from the same reasoning used to conclude the proof of Lemma 5.3. Since rd0 ðxÞAL; we must have wðgcÞAZ: From fcðgcÞ¼g; it follows that

degðcÞ degðcÞ N q pq ðÀwðgcÞÞ ¼ ÀwðgÞp : q À 1

The last inequality follows from the hypothesis and Lemma 5.3. This shows that degðcÞ is bounded. Consequently, only finitely many primes can satisfy this condition. c[ c We have assumed that d: If j d; then fd ðgcÞAG; and so x ¼ eðfcðlcÞÞ: We will show that xAL implies that fcðlcÞAL: Assume this for a moment. By the local uniform boundedness theorem, Theorem 2.5, the size of Lf½tors is bounded by ½L:K=ðqÀ1Þ q Â NM (see the remark following Proposition 4.1). Since fcðlcÞAL is a non-zero c-division point for f; only finitely many primes c can satisfy this condition. We now show that x ¼ eðfcðlcÞÞAL implies fcðlcÞAL: Let l ¼ fcðlcÞ: It is easy to see that l is separable over L: Suppose sAGalðLsep=LÞ: Then x ¼ xs implies eðlÞ¼ eðlsÞ; and so eðl À lsÞ¼0: Recall that G is torsion-free. Since l À lsAG is an c- torsion element for f; we have l ¼ ls: This is true for all sAGalðLsep=LÞ; and so lAL; as required. We note that the proof did not use the fact that c is a prime. If aAA is not zero and la is a primitive a-division point for f; then x ¼ eðlaÞAL implies laAL: We have shown that there are only finitely many primes c for which r can have an c-torsion point in L; where r is any element of DAðE; S; NÞ: If c is such a prime and r is any such Drinfeld module, we now investigate when a primitive cm-torsion point for r can be rational over L: As before, we can assume: S ¼fPg; M is a prime of E lying above P; and r has stable but bad reduction at M: ARTICLE IN PRESS

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A primitive cm-division point x for r must have the form

x ¼ eðfcðlcm Þþfd ðgcm ÞÞ:

Assuming that xAL we will deduce a bound on m: cm[ Let us assume to begin with that d: Then, we can write fd ðgcm Þ¼fbðgcmo Þ; 0 0 mo where l[b and 1pmopm: Now choose b such that bb 1 ðmod c Þ: Then, rb0 ðxÞ¼eðfb0cðlcm ÞÞ þ eðgcmo Þ and wðrb0 ðxÞÞ ¼ wðeðgcmo ÞÞ ¼ wðgcmo Þ: If xAL; so is mo degðcÞ rb0 ðxÞ from which it follows that wðgcmo ÞAZ: Since q wðgcmo Þ¼wðgÞ; we deduce (as above) that N mo degðcÞ mo degðcÞ q pq ðÀwðg mo ÞÞ ¼ ÀwðgÞp : l q À 1

This inequality bounds mo: If m ¼ mo; we are done. If moom; then l[c: Otherwise, x would not be a primitive m c m m -division point for r: Now, apply rcmo to x: We ﬁnd rcmo ðxÞ¼eðfcðlc À o ÞÞ: The left-hand side of this last equation is in L: As we have argued above, this forces m m m m c[ m m c À o fcðlc À o Þ to be in L as well. Since c; fcðlc À o Þ is a primitive -division point for f: Using Theorem 2.5 once again, we ﬁnd that m À mo is bounded. Thus, m is bounded, and we are done in the case cm[d: m c m Finally, suppose j d: Then fd ðgcm ÞAG; and so x ¼ eðfcðlc ÞÞ: This implies m m c[ m c fcðlc ÞAL: Since l j d; we must have c; and so fcðlc Þ is a primitive -division point for f: Invoking Theorem 2.5 one last time, this condition bounds m; and the proof is complete. &

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[13] B. Poonen, Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture, Math. Ann. 308 (1997) 571–586. [14] M. Rosen, Number Theory in Function Fields, GTM 210, Springer, New York, 2002. [15] A. Schweizer, On the uniform boundedness conjecture for Drinfeld modules, to appear. [16] T. Takahashi, Good reduction of elliptic modules, J. Math. Soc. Japan 34 (3) (1982) 475–487.