LOCAL HEIGHT FUNCTIONS AND THE MORDELL-WEIL THEOREM FOR DRINFELD MODULES

BJORN POONEN

January 18, 1994

Abstract. We prove an analogue for Drinfeld modules of the Mordell-Weil theorem on abelian varieties over number fields. Specifically, we show that if ϕ is a Drinfeld A-module over a finite extension L of the fraction field of A, then L considered as an A-module via ϕ is the direct sum of a free Amodule of rank ℵ_{0} with a finite torsion module. The main tool is the canonical global height function defined by Denis. By developing canonical local height functions, we are also able to show that if ϕ is defined over the ring of Sintegers O_{S} in L, then O_{S} and L/O_{S} considered as A-modules via ϕ also are each isomorphic to the direct sum of a free A-module of rank ℵ_{0} with a finite torsion module. Finally, if M is a nontrivial finite separable extension of L, then the quotient module M/L as well is isomorphic to the direct sum of a free A-module of rank ℵ_{0} with a finite torsion module.

1. Introduction

For abelian varieties over a number field, there is the well-known Mordell-Weil theorem, which states that the group of rational points is a finitely generated abelian group, and hence is isomorphic to the direct sum of its torsion subgroup with a free abelian group ℤ^{r} of some finite rank r. This paper studies the analogous question for Drinfeld modules. As usual, let K be a global function field, fix a nontrivial place ∞ of K and let A be the ring of elements of K which are integral away from ∞. If ϕ is a Drinfeld A-module defined over a finite extension L of the fraction field K of A, then L becomes an A-module via ϕ, and one can ask for a description of this A-module. (Complete definitions will be given in the next section.)

This A-module, which we call ϕ(L), is never finitely generated, as follows from the remark at the beginning of the proof of Theorem 5 in [2]. (So the most obvious analogue of the Mordell-Weil theorem is false.) Our main result is that in fact ϕ(L) is isomorphic to the direct sum of a free A-module of rank ℵ_{0} with a finite torsion module. This makes it impossible to define an interesting “Mordell-Weil rank” for Drinfeld modules based on the A-module structure of ϕ(L) alone. On the other hand, it reveals a close analogy between Drinfeld modules and the multiplicative group 𝔾_{m} over number fields, as we will discuss in Section 7.

1991 Mathematics Subject Classification. Primary 11G09.

Key words and phrases. Drinfeld modules, height functions, Mordell-Weil theorem.

The author is supported by an Office of Naval Research Fellowship.

If S is a finite nonempty set of places of L (which we identify with nonarchimedean valuations), the ring of S-integers in L is

O_{S} = { x ∈ L | v(x) ≥ 0 for all places v of L not in S }.

Any Drinfeld A-module ϕ over L can be defined over O_{S} for some S. (See Lemma 2.) Then this O_{S} becomes an A-module ϕ(O_{S}) as well, and we can show that ϕ(O_{S}) and ϕ(L)/ϕ(O_{S}) also are each isomorphic to the direct sum of a free A-module of rank ℵ_{0} with a finite torsion module. Finally, if M is a nontrivial finite separable extension of L, then ϕ(L) is a submodule of ϕ(M), and we show that ϕ(M)/ϕ(L) is the direct sum of a free A-module of rank ℵ_{0} with a finite torsion module.

The method of proof is as follows. First we develop a theory of canonical local height functions for Drinfeld modules, building on the work of Denis on canonical global height functions [3], and we use these height functions to show that the Amodules in question are of rank ℵ_{0} and are tame, meaning that every submodule of finite rank is finitely generated. (In fact, for the module structure of L alone, we only need global height functions, but for O_{S} and L/O_{S} we really need the local height functions. In any case, it seems likely that the local height functions defined here will have other applications as well.) The proof of tameness is similar to the proof of the Mordell-Weil theorem for abelian varieties given the weak Mordell-Weil theorem. To complete the proof, we classify all tame modules of rank ℵ_{0} over a Dedekind domain.

The results of this paper can undoubtedly be generalized to certain higher dimensional t-modules, probably to the same class of t-modules for which Denis [3] is able to define his canonical global height function, among others those in which the action of t as an endomorphism of 𝔾_{a}^{n} is of the form

a_{0} + a_{1}τ + ⋯ + a_{d}τ^{d}

where a_{i} ∈ M_{n}(L), τ denotes the Frobenius endomorphism on 𝔾_{a}^{n}, and a_{d} is an invertible n by n matrix. We have restricted the discussion to Drinfeld modules for simplicity.

2. Review of Drinfeld Modules

All the material of this section can be found in [6] or in Drinfeld’s original paper [4]. We use the following notation throughout our paper (except in Lemma 3 and in the appendix, where we generalize by allowing A to be any Dedekind domain and K its fraction field):

𝔽_{q} = the field of q elements (q = p^{m} for some prime p)
K = a global function field with field of constants 𝔽_{q}

(i.e., a finite extension of 𝔽_{q}(t) in which 𝔽_{q} is algebraically closed)

∞ = a fixed nontrivial place of K
A = the set of elements of K which are integral at all places except possibly ∞

(This is a Dedekind ring with field of fractions K.)

| | = the absolute value associated with ∞,

normalized so that |a| = #(A/a) for nonconstant a ∈ A Let L be an A-field, that is, a field equipped with a ring homomorphism ι : A → L. Let 𝔾_{a} be the additive group scheme over L. The ring End_{L} 𝔾_{a} of endomorphisms of 𝔾_{a} as a group scheme over L is a twisted polynomial ring L{τ} generated over L by the p^{th}-power Frobenius morphism τ, with the relation τα = α^{p}τ for all α ∈ L. Each twisted polynomial represents a polynomial map on 𝔾_{a}, and we define the degree of a twisted polynomial to be the degree of this polynomial map. (For example, the twisted polynomial 2τ^{2} + 3 represents the map x ↦ 2x^{p^{2}} + 3x, and hence deg(2τ^{2} + 3) = p^{2}.) By convention deg 0 = 0. Let D : End_{L} 𝔾_{a} → L be the map taking an endomorphism to its derivative at zero; explicitly, D : L{τ} → L takes a twisted polynomial to its constant term. Then a Drinfeld A-module over L is a ring homomorphism

ϕ : A → End_{L} 𝔾_{a} = L{τ}
a ↦ ϕ_{a}

which satisfies D ∘ ϕ = ι, and which is not the trivial homomorphism sending each a ∈ A to the constant polynomial ι(a) ∈ L{τ}. Informally, one can think of a Drinfeld A-module as the additive group of L with an A-module structure where each a ∈ A acts as a polynomial map ϕ_{a} ∈ L[x] (with a few additional conditions).

Drinfeld showed that for each Drinfeld A-module ϕ there is a unique positive integer r such that deg ϕ(a) = |a|^{r} for all a ∈ A. This integer is called the rank of ϕ. If ϕ′ is another Drinfeld A-module defined over L, a morphism from ϕ to ϕ′ is an element u ∈ End_{L} 𝔾_{a} such that uϕ_{a} = ϕ′_{a}u for all a ∈ A. By taking degrees, we see that if a nonzero morphism from ϕ to ϕ′ exists, then ϕ and ϕ′ have the same rank.

3. The Canonical Local Function Associated with a Drinfeld Module

Let L be an A-field which is also a local field; i.e., it is complete with respect to a discrete valuation v and the residue field is finite. For x ∈ L, let ṽ(x) = min{0, v(x)}.

Lemma 1. If d > 0 and f(x) = c_{d}x^{d} + c_{d−1}x^{d−1} + ⋯ + c_{0} ∈ L[x], then ṽ(f(x)) − dṽ(x) is bounded. Also ṽ(f(x)) = dṽ(x) + v(c_{d}) when ṽ(x) is sufficiently negative.

Proof. If v(x) is sufficiently negative, then v(c_{d}x^{d}) < v(c_{i}x^{i}) for all i < d, so

v(f(x)) = v(c_{d}x^{d}) = dv(x) + v(c_{d}),

which is negative if v(x) is sufficiently negative. For the other x, the lower bound on v(x) gives a lower bound on v(f(x)) by the triangle inequality.

Let ϕ be a Drinfeld A-module over L. Fix a ∈ A \ 𝔽_{q}. (We will later show that the choice of a is inconsequential.) Then |a| > 1, since otherwise a would be integral at all places (including infinity), forcing it to be in 𝔽_{q}.

The next proposition defines a nonpositive function V : L → ℝ associated with ϕ, which behaves like a canonical local height function (except that it is nonpositive, so in the next section we scale it by a negative constant). The definition is modeled on Tate’s definition of the (Néron-Tate) canonical global height function for abelian varieties, as was Denis’ definition of the canonical global height function for a Drinfeld module.

Proposition 1.

(1) For x ∈ L, the limit

V (x) = lim_{n→∞}(deg ϕ_{a^{n}})^{−1}ṽ(ϕ_{a^{n}}(x))

exists.

(2) V (x) − ṽ(x) is bounded.

(3) If v(x) is sufficiently negative, then V (x) = v(x) + v(c)/(d − 1), where d = deg ϕ_{a} and c is the leading coefficient of ϕ_{a} considered as a polynomial map.

(4) V (x ± y) ≥ min{V (x), V (y)} and V (−x) = V (x).

(5) If x_{1}, …, x_{n} ∈ L and there is an i such that V (x_{i}) < V (x_{j}) for all j ≠ i, then x_{1} + ⋯ + x_{n} ≠ 0.

Proof. First note that d = |a|^{r} > 1, where r is the rank of ϕ, and deg ϕ_{a^{n}} = d^{n}. Let M be the bound on v̂(ϕ_{a}(x)) − dṽ(x) given by Lemma 1. Then

|(deg ϕ_{a^{n+1}})^{−1}ṽ(ϕ_{a^{n+1}}(x)) − (deg ϕ_{a^{n}})^{−1}ṽ(ϕ_{a^{n}}(x))|
= d^{−(n+1)}|ṽ(ϕ_{a}(ϕ_{a^{n}}(x))) − dṽ(ϕ_{a^{n}}(x))|
≤ d^{−(n+1)}M.

Thus the limit in (1) exists, and (2) follows as well, since ∑_{n=0}^{∞} d^{−(n+1)}M is finite.

If v(x) is sufficiently negative, then by Lemma 1 and induction on n we see that for all n ≥ 0, ṽ(ϕ_{a^{n}}(x)) is at least as negative as v(x), and that

ṽ(ϕ_{a^{n+1}}(x)) = dṽ(ϕ_{a^{n}}(x)) + v(c).

This can be rewritten

ṽ(ϕ_{a^{n+1}}(x))/d^{n+1} = ṽ(ϕ_{a^{n}}(x))/d^{n} + v(c)/d^{n+1},

so

ṽ(ϕ_{a^{N}} (x))/d^{N} = ṽ(ϕ_{a^{0}}(x))/d^{0} + ∑_{i=1}^{N} v(c)/d^{i}

= v(x) + ∑_{i=1}^{N} v(c)/d^{i}.

Letting N → ∞, and summing the infinite geometric series gives (3).

The first half of (4) follows by applying min{0, ·} to

v(ϕ_{a^{n}}(x ± y)) = v(ϕ_{a^{n}}(x) ± ϕ_{a^{n}}(y)) ≥ min{v(ϕ_{a^{n}}(x)), v(ϕ_{a^{n}}(y))},

dividing by deg ϕ_{a^{n}} and letting n → ∞. The second half of (4) follows in the same way.

Finally, (5) is proved from (4) in the same way that is it proved for valuations.

The usefulness of V springs from the following.

Proposition 2. Let ϕ′ be another Drinfeld A-module over L, with corresponding function V′ obtained from Proposition 1, and let u : ϕ → ϕ′ be a morphism of Drinfeld modules. Then for all x ∈ L,

V′(u(x)) = (deg u)V (x).

Proof. If u = 0, both sides are zero, so assume u ≠ 0. Then ϕ and ϕ′ have the same rank r, and deg ϕ_{a^{n}} = |a^{n}|^{r} = deg ϕ′_{a^{n}}. Let M be the bound on ṽ(u(x))−(deg u)ṽ(x) given by Lemma 1. Then

|ṽ(ϕ′_{a^{n}}(u(x))) − (deg u)ṽ(ϕ_{a^{n}}(x))| = |ṽ(u(ϕ_{a^{n}}(x))) − (deg u)ṽ(ϕ_{a^{n}}(x))|
≤ M.

Now divide by deg ϕ_{a^{n}} (which equals deg ϕ′_{a^{n}}), and let n → ∞ to get

|V′(u(x)) − (deg u)V (x)| ≤ 0.

Corollary 1. If b ∈ A,

V (ϕ_{b}(x)) = (deg ϕ_{b})V (x)

for all x ∈ L.

Proposition 3. V is independent of the choice of a ∈ A \ 𝔽_{q}.

Proof. Let V be defined as before using a ∈ A\𝔽_{q}, and suppose b is another element of A \ 𝔽_{q}. Since deg ϕ_{b^{n}} → ∞ as n → ∞, and V − ṽ is bounded, we have

lim_{n→∞}(deg ϕ_{b^{n}})^{−1}ṽ(ϕ_{b^{n}}(x)) = lim_{n→∞}(deg ϕ_{b^{n}})^{−1}V (ϕ_{b^{n}}(x))
= V (x)

by Corollary 1.

Let O = { x ∈ L | v(x) ≥ 0 } be the valuation ring of L, let π ∈ O be a uniformizing parameter (i.e., v(π) = 1), and let l = O/π be the residue field. We say that the Drinfeld module ϕ is defined over O if for each a ∈ A, all coefficients of ϕ_{a} belong to O.

Proposition 4. Suppose ϕ is defined over O. Then

(1) V (x + y) = V (x) whenever x ∈ L and y ∈ O. In other words, V induces a function on L/O.

(2) If C is a real constant, only finitely many elements x̄ of L/O satisfy V (x̄) ≥ C.

(3) V (x) = 0 if and only if ϕ_{b}(x) ∈ O for some nonzero b ∈ A.

(4) If in addition, for some a ∈ A \ 𝔽_{q}, the leading coefficient of ϕ_{a} is a unit of O, then V (x) = ṽ(x) for all x ∈ L.

Proof. If y ∈ O, then ϕ_{a^{n}}(y) ∈ O, so by definition of V, V (y) = 0. By (4) in Proposition 1,

V (x + y) ≥ min{ V (x), V (y) } = V (x),

since V (x) ≤ 0 for any x ∈ L. The same argument with x and y replaced by x + y and −y shows V (x) ≥ V (x + y), so V (x + y) = V (x), proving (1).

Now V and ṽ are both functions defined on L/O and they differ by a bounded amount by (2) in Proposition 1, so to prove part (2), it suffices to show that for each constant C, there are finitely many x̄ ∈ L/O such that ṽ(x̄) ≥ C. This is equivalent to showing that π^{−n}O/O is finite for each n ≥ 1. This is finite, because it has a composition series as an O-module

π^{−n}O/O ⊃ π^{−(n−1)}O/O ⊃ ⋯ O/O,

in which each quotient is isomorphic to the residue field l, which was assumed to be finite. This proves (2).

If ϕ_{b}(x) ∈ O for some nonzero b ∈ A, then V (ϕ_{b}(x)) = 0. On the other hand, by Corollary 1, V (ϕ_{b}(x)) = (deg ϕ_{b})V (x) and deg ϕ_{b} ≠ 0 since ϕ is injective (see Proposition 2.1 in [4]), so V (x) = 0. Conversely, if V (x) = 0, then V (ϕ_{b}(x)) = (deg ϕ_{b})V (x) = 0 for all b ∈ A. But A is infinite, and { y ∈ L | V (y) = 0 } consists of only finitely many cosets of O, so some ϕ_{b}(x) and ϕ_{b′}(x) belong to the same coset. Then ϕ_{b−b′}(x) ∈ O. This proves (3).

Finally, to prove (4), notice that for any polynomial ϕ_{a} ∈ O[t] whose leading coefficient is a unit,

ṽ(ϕ_{a}(x)) = (deg ϕ_{a})ṽ(x)

for all x ∈ L. Hence by induction on n,

ṽ(ϕ_{a_{n}}(x)) = (deg ϕ_{a_{n}})ṽ(x)

for all x ∈ L. Divide by deg ϕ_{a_{n}} and let n → ∞ to get V (x) = ṽ(x).

4. Local and Global Heights

For the next three sections, L will be a finite extension of K. We make L an A-field using the inclusion maps A ⊂ K ⊂ L. Let ϕ be a Drinfeld A-module over L.

Let v be a place of L (by place, we mean a nontrivial place). We normalize the valuation v to take values in ℤ ∪ ∞. The completion L_{v} of L at v is a local field, and we can consider ϕ as a Drinfeld A-module over L_{v}. In particular, for each v we get a function V_{v} as in the previous section. We define the canonical local height on L associated with ϕ and v to be the real valued function

ĥ_{v}(x) = −[L : K]^{−1}d(v)V_{v}(x),

where d(v) is the degree of the residue field of v over 𝔽_{q}. Since ĥ_{v} is simply a constant multiple of V_{v}, all the results of the previous section concerning V can be translated into results about ĥ_{v}.

On the other hand, Denis has defined a canonical global height function ĥ associated with ϕ. (Actually, his definition is given for certain higher dimensional t-modules as well, but for the case A = 𝔽_{q}[t] only. There is no problem with extending the definition to other A.) If a ∈ A \ 𝔽_{q}, then (a restatement of) his definition is

ĥ(x) = lim_{n→∞}(deg ϕ_{a^{n}})^{−1}h(ϕ_{a^{n}}(x)),

where h(x) denotes the Weil height on 𝔸^{1}(K̄). (If x belongs to a finite extension E of K, then

h(x) = −[E : K]^{−1} ∑_{places w of E} d(w) min{0, w(x)},

where d(w) denotes the degree of the residue field of w over 𝔽_{q}, and w(x) is normalized to take values in ℤ ∪ ∞.)

We recall some properties (due to Denis) of this global height function for later use.

Proposition 5.

(1) If a ∈ A, and x ∈ K̄, then

ĥ(ϕ_{a}(x)) = (deg ϕ_{a})ĥ(x).

(2) If C is a real constant, there are only finitely many x ∈ L with ĥ(x) ≤ C.

(3) ĥ(x) = 0 if and only if ϕ_{a}(x) = 0 for some nonzero a ∈ A. By the previous statement, the number of such x belonging to L is finite.

(4) ĥ(x ± y) ≤ ĥ(x) + ĥ(y).

Proof. See [3].

The last goal of this section is to relate our local height functions with Denis’ global height function. The following lemma will allow us to replace sums over all places of L with sums over a finite number of places.

Lemma 2. A Drinfeld A-module ϕ over L is defined over the valuation ring O_{v} corresponding to v for all but finitely many places v. In other words, ϕ is defined over the ring of S-integers O_{S} for some finite set of places S.

Proof. The ring A is finitely generated. (In fact, if a ∈ A \ 𝔽_{q}, then A is a finitely generated module over 𝔽_{q}[a].) If a_{1}, …, a_{n} are a set of generators, and v corresponds to a prime not occurring in the denominators of ϕ_{a_{1}}, …, ϕ_{a_{n}}, then that prime will not occur in the denominators of ϕ_{a} for any a ∈ A, since ϕ_{a} can be expressed as a sum of compositions of the ϕ_{a_{i}}. This holds for all but finitely many v.

Proposition 6. If x ∈ L, then ĥ_{v}(x) = 0 for all but finitely many places v of L, and

ĥ(x) = ∑_{places v of L} ĥ_{v}(x).

Proof. Let S be the set of places v such that either v(x) < 0 or ϕ is not definable over the corresponding valuation ring. By Lemma 2, S is finite. If v ∉ S, then Proposition 4 shows V_{v}(x) = 0 and hence ĥ_{v}(x) = 0, proving the first claim.

Now v(ϕ_{a^{n}}(x)) ≥ 0 if v ∉ S, so

h(ϕ_{a^{n}}(x)) = −[L : K]^{−1} ∑_{v∈S} d(v) min{0, v(ϕ_{a^{n}}(x))}.

If we now divide by deg ϕ_{a^{n}} and let n → ∞, we obtain

ĥ(x) = −[L : K]^{−1} ∑_{v∈S} d(v)V_{v}(x)
= ∑_{v∈S} ĥ_{v}(x)
= ∑_{all places v of K} ĥ_{v}(x),

since we just showed that ĥ_{v}(x) = 0 when v ∉ S.

5. A Mordell-Weil Type Theorem

For the next two sections we retain the assumption that ϕ is a Drinfeld A-module over a finite extension L of K. The additive group of L becomes an A-module by letting each a ∈ A act as the polynomial map ϕ_{a}. We will use the notation ϕ(L) to denote this A-module. Our goal is to characterize ϕ(L) (as an A-module).

By the rank of an A-module M we mean the dimension of the K-vector space M ⊗_{A} K, which is some cardinal number. The torsion submodule of M is

M_{tors} = { x ∈ M | ax = 0 for some a ∈ A },

which is also the kernel of the natural map M → M ⊗_{A} K. An A-module is tame if every submodule of finite rank is finitely generated as an A-module. We will show that ϕ(L) is not finitely generated, but at least it is tame.

Some parts of the proof are similar to the proof of the Mordell-Weil theorem for abelian varieties. What makes a direct reproduction of the argument impossible is the failure of the “weak Mordell-Weil theorem,” which for abelian varieties says that if G is the group of rational points and n ≥ 1, then G/nG is finite. To prove the tameness in our situation, we use the following as a substitute.

Lemma 3. Let A be any Dedekind domain, let M be an A-module of finite rank, and suppose the torsion submodule M_{tors} is a finitely generated A-module. Then for any nonzero ideal I ⊂ A, M/IM is a finitely generated A-module. In particular, if a is a nonzero element of A and A/a is finite, then the finitely generated A/a-module M/aM must be finite.

Proof. If M′ is the image of M in M ⊗_{A} K, then we have an exact sequence

0 → M_{tors} → M → M′ → 0.

Tensoring with A/I yields a right exact sequence

M_{tors}/IM_{tors} → M/IM → M′/IM′ → 0,

and M_{tors}/IM_{tors} is finitely generated, so it suffices to show M′/IM′ is finitely generated. Hence without loss of generality we may assume M is a sub-A-module of K^{r} for some r ≥ 0.

If we knew that for given nonzero ideals I, J, the result held for all M, then for any sub-A-module M ⊂ K^{r}, M/JM and JM/I(JM) would be finitely generated, so M/IJM would be finitely generated too. Thus we can reduce to the case where I is a nonzero prime ideal 𝔭 of A.

In fact we claim that dim_{A/𝔭} M/𝔭M ≤ r. It suffices to show that if m_{1}, …, m_{n} are elements of M whose images in M/𝔭M are independent over A/𝔭, then m_{1}, …, m_{n} are independent over K as well. Suppose not; i.e., suppose that α_{1}m_{1} +⋯ α_{n}m_{n} = 0 for some α_{1}, …, α_{n} ∈ K. By multiplying by some power of a uniformizing parameter for 𝔭, we may assume that v(α_{i}) ≥ 0 for all i, with equality for at least one i. Then reduction modulo 𝔭 shows that the images of m_{1}, …, m_{n} are not independent over A/𝔭.

Thus M/𝔭M can be generated by less than or equal to r elements as an A/𝔭vector space, or equivalently as an A-module, as desired.

Lemma 4. ϕ(L) is a tame A-module.

Proof. Suppose M is a submodule of ϕ(L) of finite rank. We must show M is finitely generated. By (3) in Proposition 5, the torsion submodule of ϕ(L) is finite, so the same is true for M. Thus we may apply Lemma 3 to deduce that M/aM is finite for any nonzero a ∈ A.

Pick a ∈ A \ 𝔽_{q}. Then deg ϕ_{a} ≥ 2. Let S ⊂ M be a set of representatives for M/aM. Let C = max_{s∈S} ĥ(s). Let T be the union of S with { x ∈ L | ĥ(x) ≤ C }. By (2) in Proposition 5, T is finite.

We claim that T generates M as an A-module. Let N be the submodule generated by T. If N ≠ M, then by (2) in Proposition 5, we can pick m_{0} ∈ M \ N with ĥ(m_{0}) minimal. Since S is a set of representatives for M/aM, we can write m_{0} = s + ϕ_{a}(m) for some s ∈ S and m ∈ M. Moreover m ∉ N, since otherwise m_{0} ∈ N also. Then

2ĥ(m_{0}) ≤ 2ĥ(m)  (by the minimality of m_{0})
≤ (deg ϕ_{a})ĥ(m)
= ĥ(ϕ_{a}(m))  (by (1) in Proposition 5)
= ĥ(m_{0} − s)
≤ ĥ(m_{0}) + ĥ(s)  (by (4) in Proposition 5)
≤ ĥ(m_{0}) + C

so ĥ(m_{0}) ≤ C, and m_{0} ∈ T ⊂ N, contradicting the definition of m_{0}. Thus M is finitely generated, as desired.

Theorem 1. The A-module ϕ(L) is the direct sum of its torsion submodule, which is finite, with a free A-module of rank ℵ_{0}.

Proof. First we compute the rank of ϕ(L). Since L is countable even as a set, the rank of ϕ(L) is at most ℵ_{0}. Suppose the rank is not ℵ_{0}; i.e., suppose it is finite. Then the tameness (Lemma 4) implies that ϕ(L) is finitely generated. Let Z be a finite set of generators. We can find a finite set S of places of L such that ϕ is defined over O_{S} (by Lemma 2), and which is large enough that Z ⊂ O_{S}. Then O_{S} is a proper submodule over ϕ(L) containing Z, contradicting the fact that Z generates all of ϕ(L).

Thus the rank of ϕ(L) is ℵ_{0}. Now applying Proposition 10 from the appendix yields the desired result. (The finiteness of the torsion submodule is (3) in Proposition 5.)

There is nothing mysterious about the finite torsion submodule. For a given ϕ and L, it can be calculated effectively by bounding the Weil height of a torsion point, or by using reductions modulo various primes of L. By Theorem 10.15 in [7], every finitely generated torsion module M over a Dedekind domain A is isomorphic to a direct sum

A/I_{1} ⊕ ⋯ ⊕ A/I_{n}

where I_{1} ⊂ ⋯ ⊂ I_{n} are nonzero ideals in A. Moreover, the I_{j} are uniquely determined by M. In fact, I_{j} is the annihilator of the j^{th} exterior power of M, as pointed out by Michael Rosen.

We conclude this section by remarking that the Drinfeld module structures of the algebraic closure L̄ and separable closure L^{sep} are much easier to determine.

Proposition 7. Each of the A-modules ϕ(L̄) and ϕ(L^{sep}) is the direct sum of a K-vector space of dimension ℵ_{0} with a torsion module isomorphic to (K/A)^{r}, where r is the rank of ϕ.

Proof. If a is a nonzero element of A and c ∈ L̄, then the equation ϕ_{a}(x) = c can be solved in L̄. Moreover, if c ∈ L^{sep}, then any solution x_{0} lies in L^{sep}, because the polynomial ϕ_{a}(x) − c is separable (since its derivative is the nonzero constant a). This means that ϕ(L̄) and ϕ(L^{sep}) are divisible A-modules.

By Theorem 7 in [10], any divisible A-module is the direct sum of a K-vector space and the torsion submodule. The torsion submodule in both of our cases is (K/A)^{r} by Proposition 2.2 in [4]. The K-vector space must have dimension at least ℵ_{0}, because even ϕ(L) has rank ℵ_{0}, by Theorem 1. On the other hand, both L̄ and L^{sep} are countable even as sets, so the vector space must have dimension exactly ℵ_{0}.

6. Other Module Structure Theorems

In this section we answer some related questions about modules arising from Drinfeld modules. (These were posed by David Goss.) Fix a finite set S of places of L such that ϕ is defined over the ring of integers O_{S}. (The existence of S is guaranteed by Lemma 2.) Then each polynomial map ϕ_{a} maps O_{S} into O_{S}, so we get a submodule ϕ(O_{S}) of ϕ(L). We want to describe the A-module structures of ϕ(O_{S}) and ϕ(L)/ϕ(O_{S}).

For each place v ∉ S, the inclusion of L in its completion L_{v} induces a group homomorphism L/O_{S} → L_{v}/O_{v} where O_{v} = { x ∈ L_{v} | v(x) ≥ 0 }, and as is well known, the map

L/O_{S} → ⊕_{v∉S}L_{v}/O_{v}

is an isomorphism. The Drinfeld module ϕ induces module structures on L_{v} and O_{v} as well, and it is clear that the isomorphism above respects these module structures. So we can understand ϕ(L)/ϕ(O_{S}) by understanding ϕ(L_{v})/ϕ(O_{v}) for each v ∉ S.

Lemma 5. The A-modules ϕ(O_{S}) and ϕ(L_{v})/ϕ(O_{v}) (for v ∉ S) are tame.

Proof. A submodule of a tame module is tame, so the tameness of ϕ(O_{S}) follows from Lemma 4.

The proof of the tameness of ϕ(L_{v})/ϕ(O_{v}) is the same as the proof of the tameness of ϕ(L), except using the function V_{v} (or ĥ_{v}) defined by Proposition 1, instead of ĥ. In particular we use Corollary 1 instead of (1) in Proposition 5, (2) in Proposition 4 instead of (2) in Proposition 5, Proposition 4 instead of (3) in Proposition 5 to show that the torsion submodule of ϕ(L_{v})/ϕ(O_{v}) is finite, and (4) in Proposition 1 instead of (4) in Proposition 5.

Computing the ranks of these A-modules is now more difficult than for ϕ(L). We will use the following simple lemma.

Lemma 6. Suppose p is a prime number and α ∈ ℚ. Then for each k ≥ 1, there exist positive integers n_{1}, …, n_{k} such that the rational numbers p^{i}(α − n_{j}) for i ranging over nonnegative integers and 1 ≤ j ≤ k are all distinct and negative.

Proof. Choose n_{1}, …, n_{k} to be large consecutive integers. Then each ratio (α − n_{j})/(α − n_{j′}) will be close to 1, and in particular will not be a power of p.

Lemma 7. The A-modules ϕ(O_{S}) and ϕ(L_{v})/ϕ(O_{v}) (for v ∉ S) have rank ℵ_{0}.

Proof. First of all, the set O_{S} is countable since O_{S} ⊂ L, and L_{v}/O_{v} is countable as well since it is a direct summand of L/O_{S}, and L is countable. Hence the ranks of ϕ(O_{S}) and ϕ(L_{v})/ϕ(O_{v}) are at most ℵ_{0}. It will suffice in each case to exhibit for each k ≥ 1 elements x_{1}, …, x_{k} of the module which are A-independent.

Let us consider ϕ(O_{S}) first. Pick a place w in S, and let V be the corresponding local function. Then w(O_{S}) contains all sufficiently negative integers by the Riemann-Roch theorem, so by (3) in Proposition 1, there exists (a very negative) α ∈ ℚ such that V (O_{S}) contains α − n for every positive integer n. Choose n_{1}, …, n_{k} as in Lemma 6, and choose x_{i} ∈ O_{S} such that V (x_{i}) = α − n_{i}.

We claim that x_{1}, …, x_{k} are A-independent in ϕ(O_{S}). Suppose not. Then for some a_{1}, …, a_{k} ∈ A,

ϕ_{a_{1}}(x_{1}) + ⋯ + ϕ_{a_{k}}(x_{k}) = 0.

But

V (ϕ_{a_{i}}(x_{i})) = (deg ϕ_{a_{i}})V (x_{i}) by Corollary 1
= (deg ϕ_{a_{i}})(α − n_{i}),

and deg ϕ_{a_{i}} is 0 or a power of p, so by the choice of n_{i}, the values V (ϕ_{a_{i}}(x_{i})) are all distinct and negative, after one throws out the i for which a_{i} = 0. By (5) in Proposition 1, this forces a_{i} = 0 for all i. Thus x_{1}, …, x_{k} are A-independent, as desired.

We use virtually the same construction to find independent x_{1}, …, x_{k} in ϕ(L_{v})/ϕ(O_{v}). This time let V be the local function corresponding to v. As before, we can pick α ∈ ℚ such that V (L_{v}) contains α−n for every positive integer n. Choose n_{1}, …, n_{k} as in Lemma 6, and choose x_{i} ∈ L_{v} such that V (x_{i}) = α − n_{i}. Then the images of x_{1}, …, x_{k} in ϕ(L_{v})/ϕ(O_{v}) are A-independent for the same reason as before.

Lemma 8. For all but finitely many places v, the A-module ϕ(L_{v})/ϕ(O_{v}) is torsionfree.

Proof. Pick a ∈ A \ 𝔽_{q}, and let V_{v} be the function on the local field L_{v} defined by Proposition 1. By Lemma 2, for all but finitely many places v, ϕ is defined over O_{v} and the leading coefficient of ϕ_{a} is a unit at v. By (3) and (4) in Proposition 4, for any such v, ϕ(L_{v})/ϕ(O_{v}) is torsion-free.

Theorem 2. Each of the A-modules ϕ(O_{S}), ϕ(L)/ϕ(O_{S}), and ϕ(L_{v})/ϕ(O_{v}), is the direct sum of a free A-module of rank ℵ_{0} and a finite torsion module.

Proof. For ϕ(O_{S}) and ϕ(L_{v})/ϕ(O_{v}), combine Lemmas 5 and 7 with Proposition 10 to obtain the result. (The finiteness of the torsion submodule of ϕ(O_{S}) follows from (3) in Proposition 5, and the finiteness of the torsion submodule of ϕ(L_{v})/ϕ(O_{v}) follows from Proposition 4.)

Now ϕ(L)/ϕ(O_{S}) is the countable direct sum of modules ϕ(L_{v})/ϕ(O_{v}), each isomorphic to the direct sum of a free module of rank ℵ_{0} and a finite torsion module, so ϕ(L)/ϕ(O_{S}) is the direct sum of a free A-module of rank ℵ_{0} · ℵ_{0} = ℵ_{0} and a torsion module. This torsion module is finite as well, because of Proposition 8.

Next we describe the A-module structure of ϕ(M)/ϕ(L) where M is a finite extension of L.

Lemma 9. If M is a finite separable extension of L, then ϕ(M)/ϕ(L) is tame.

Proof. Since a submodule of a tame module is tame, we may without loss of generality enlarge M to assume M is a Galois extension of L. Let σ_{1}, …, σ_{n} be all the elements of Gal(M/L). Each σ_{i} commutes with the action of A, so we have a homomorphism of A-modules

ϕ(M) → ϕ(M) ⊕ ⋯ ⊕ ϕ(M)
x ↦ (σ_{1}x − x, …, σ_{n}x − x)

whose kernel is ϕ(L) by Galois theory. Now ϕ(M)/ϕ(L) is isomorphic to the image, which is a submodule of a finite direct sum of tame modules, and is hence tame.

Lemma 10. If M is a nontrivial finite separable extension of L, then ϕ(M)/ϕ(L) has rank ℵ_{0}.

Proof. Since we know from Theorem 1 that ϕ(M) has rank ℵ_{0}, the rank of ϕ(M)/ϕ(L) is at most ℵ_{0}. So it will suffice to construct for each k ≥ 1, elements x_{1}, …, x_{k} ∈ ϕ(M) whose images in ϕ(M)/ϕ(L) are A-independent.

Using Lemma 2, choose a finite set of places S of L such that ϕ is defined over O_{S}. Pick a ∈ A \ 𝔽_{q} and enlarge S if necessary so that the leading coefficient of ϕ_{a} is a unit at all places outside S. Then by (4) in Proposition 4, if w is any place of M lying above a place of L not in S, the local function V_{w} on M equals w̃.

By the Čebotarev Density Theorem for function fields [8], there exist primes 𝔭_{1}, …, 𝔭_{k} of L outside S which split completely in M. For 1 ≤ i ≤ k, let 𝔮_{i} and 𝔯_{i} be two primes of M above 𝔭_{i}. Next, for each i, use an approximation theorem to find x_{i} ∈ M such that x_{i} is integral at all the 𝔮_{j} and 𝔯_{j} except 𝔮_{i}, where it is not integral.

Suppose we have a dependence relation for the images of x_{1}, …, x_{k} in ϕ(M)/ϕ(L); i.e., there exist a_{1}, …, a_{k} ∈ A and y ∈ L not all zero such that

ϕ_{a_{1}}(x_{1}) + ⋯ + ϕ_{a_{k}}(x_{k}) = y.

Since everything on the left side is integral at 𝔯_{i}, so is y. Since y ∈ L, this is the same as saying y is integral at 𝔭_{i}, so y is integral at 𝔮_{i} as well. Now everything in the dependence equation is integral at 𝔮_{i} except possibly ϕ_{a_{i}}(x_{i}), so this is integral also. In other words, if w and V_{w} are the valuation and local function on M corresponding to 𝔮_{i}, then since V_{w} = w̃, we get V_{w}(ϕ_{a_{i}}(x_{i})) = 0, whereas V_{w}(x_{i}) = w̃(x_{i}) < 0. By Corollary 1, this forces a_{i} = 0. This holds for each i, so the images of x_{1}, …, x_{k} in ϕ(M)/ϕ(L) are A-independent, as desired.

Theorem 3. If M is a nontrivial finite separable extension of L, then ϕ(M)/ϕ(L) is the direct sum of a free A-module of rank ℵ_{0} and a finite torsion module.

Proof. Lemmas 9 and 10 allow us to apply Proposition 10 from the appendix. The torsion submodule is finitely generated (since it is tame), and hence finite, since all ideals of A have finite index.

Remark. Theorem 3 fails miserably if we do not require M to be separable over L. For example, if M = L^{1/p}, which is a purely inseparable extension of L of degree p, the A-module structure of ϕ(M)/ϕ(L) is the same as the usual A-module structure of M/L, because any positive power of the Frobenius τ acts as zero on M/L, so that a twisted polynomial a_{0} + a_{1}τ + ⋯ + a_{d}τ^{d} acts only by its constant term. And of course, in the usual A-module structure, M/L is simply a finite-dimensional vector space over the quotient field K of A.

7. Comparison with 𝔾_{m} over Number Fields

(Most of the material of this section is due to David Goss.) Let F be a number field with ring of integers O := O_{F}. Then the structure of F^{∗} as an abstract group (ℤ-module) is exactly analogous to the A-module structure of L given by a Drinfeld A-module over L, as described by Theorem 1. Specifically, there is the following result, which is well-known although there seems to be no good reference.

Proposition 8. As an abstract abelian group, F^{∗} is isomorphic to the product of its torsion subgroup (i.e., the finite group of roots of unity in F) and a free abelian group of rank ℵ_{0}.

Proof. Let 𝔓 be the group of principal fractional ideals. This is a subgroup of finite index in the free abelian group of divisors of F. Thus 𝔓 also is free of rank ℵ_{0}. Now let U be the group of units of O. Then there is the standard exact sequence:

0 → U → F^{∗} → 𝔓 → 0 .

The freeness of the group on the right now tells us that this sequence splits. Thus

F^{∗} ≃ U × 𝔓 .

The result now follows from Dirichlet’s unit theorem.

An obvious question is whether one can also obtain a proof of Theorem 1 in the same fashion as Proposition 8. This would be amazing since it would entail finding the analogue of the divisor group in the Drinfeld theory as well as the class group perhaps. It would also entail finding an analogue of the unit group, and thus (hopefully) a canonical finitely generated submodule of the rational points, which would again be very important.

We can state analogues of Theorems 2 and 3 for the multiplicative groups of number fields as well, although sometimes the results are not entirely similar. (Again, these may be well-known.)

Proposition 9. Let S be a finite set of places of F including the infinite ones, and let O_{S} be the ring of S-integers in F.

(1) O_{S}^{∗} is a finitely generated abelian group.

(2) F^{∗}/O_{S}^{∗} is a free abelian group of rank ℵ_{0}.

(3) If E is a finite extension of F, then E^{∗}/F^{∗} is isomorphic to the product of a free abelian group of rank ℵ_{0} with a finite torsion group.

Sketch of Proof. Part (1) is the Dirichlet S-unit theorem. Part (2) follows by consideration of divisors, as in the proof of Proposition 8. Part (3) can be proved in the same way as Theorem 3. (For the proof analogous to that of Lemma 10, choose primes 𝔭_{i} of F which split completely in E, let 𝔮_{i}, 𝔯_{i} be two primes of E above 𝔭_{i}, and choose x_{i} ∈ E^{∗} to be a unit at all the 𝔮_{j} and 𝔯_{j}, except not at 𝔮_{i}. Then considering the valuations corresponding to 𝔮_{i} and 𝔯_{i} shows that

x_{1}^{n_{1}}x_{2}^{n_{2}} ⋯ x_{k}^{n_{k}} ∈ F^{∗}

only if each integer n_{i} is zero.)

In fact, the proof of Theorem 3 was inspired by the corresponding proof for multiplicative groups of number fields rather than the other way around.

Appendix: Classification of Tame Modules of rank ℵ_{0} over a Dedekind Domain

Throughout this section, A is an arbitrary Dedekind domain, and K is its field of fractions. Recall that the rank of an A-module M is the dimension of the K-vector space M ⊗_{A} K, and that an A-module is called tame if every submodule of finite rank is finitely generated as an A-module. The goal of this section is to prove the following, which can be considered an extension of the classification theorem for finitely generated modules over a Dedekind domain.

Proposition 10. Every tame A-module M of rank ℵ_{0} is isomorphic to the direct sum of its torsion submodule M_{tors} with a free A-module of rank ℵ_{0}.

The special case where A is a principal ideal domain and M is torsion-free occurs as Exercise 52 in [9]. The only difficulty that arises from generalizing to Dedekind domains is the elimination of non-principal fractional ideals, which is handled by Lemma 12. This lemma is well known, and in fact much more general results are known [1]. Nevertheless, for the case at hand, there is a beautiful proof which is not long, so we will give it anyway.

Lemma 11. If M is a torsion-free tame A-module of rank ℵ_{0}, then

M ≅ I_{1} ⊕ I_{2} ⊕ ⋯

where each I_{i} is a projective A-module of rank 1 (i.e., a fractional ideal).

Proof. Let

0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⋯

be a full flag of K-vector spaces in M ⊗_{A} K; i.e. dim V_{i} = i and ⋃_{i=1}^{∞} V_{i} = M ⊗_{A} K. Let M_{i} = M ∩ V_{i}, so M_{i} is a finitely generated A-module with M_{i} ⊗_{A} K = V_{i} and ⋃_{i=1}^{∞} M_{i} = M. It suffices to construct for each i ≥ 1 a projective A-module I_{i} ⊆ M_{i} of rank 1 such that M_{i} = M_{i−1} ⊕ I_{i}.

We have an injection

M_{i}/M_{i−1} ↪ V_{i}/V_{i−1} ≅ K.

Here M_{i}/M_{i−1} is embedded as a finitely generated sub-A-module of K, and it is nonzero (since M_{i} ⊗_{A} K = V_{i}), so M_{i}/M_{i−1} is a projective A-module of rank 1. Since it is projective, the exact sequence

0 → M_{i−1} → M_{i} → M_{i}/M_{i−1} → 0

splits to give M_{i} ≅ M_{i−1} ⊕ I_{i}, where I_{i} is a submodule of M_{i} projecting isomorphically onto M_{i}/M_{i−1}. Thus

M = ⋃_{i=1}^{∞} M_{i} = I_{1} ⊕ I_{2} ⊕ · · · .

Lemma 12. If I_{1}, I_{2}, … are fractional ideals of A, then

I_{1} ⊕ I_{2} ⊕ ⋯ ≅ A ⊕ A ⊕ ⋯

as A-modules.

Proof. Since the isomorphism type of the direct sum of two fractional ideals I, J is determined by the ideal class of IJ, we may replace I_{2} ⊕ I_{3} on the left by the isomorphic A-module I_{1}^{−1} ⊕J_{1} where J_{1} = I_{1}I_{2}I_{3} (in the group of fractional ideals). Similarly replace I_{4} ⊕ I_{5} with J_{1}^{−1} ⊕ J_{2} where J_{2} = J_{1}I_{4}I_{5}, replace I_{6} ⊕ I_{7} with J_{2}^{−1} ⊕ J_{3} where J_{3} = J_{2}I_{6}I_{7}, etc. We get

I_{1} ⊕ (I_{2} ⊕ I_{3}) ⊕ (I_{4} ⊕ I_{5}) ⊕ (I_{6} ⊕ I_{7}) ⊕ ⋯
≅ I_{1} ⊕ (I_{1}^{−1} ⊕ J_{1}) ⊕ (J_{1}^{−1} ⊕ J_{2}) ⊕ (J_{2}^{−1} ⊕ J_{3}) ⊕ ⋯
≅ (A ⊕ A) ⊕ (A ⊕ A) ⊕ (A ⊕ A) ⊕ ⋯

as desired.

Proof of Proposition 10. If M is a tame A-module of rank ℵ_{0} then M/M_{tors} is a torsion-free A-module of rank ℵ_{0}, so by Lemmas 11 and 12, M/M_{tors} is a free A-module of rank ℵ_{0}. Because it is free, the exact sequence

0 → M_{tors} → M → M/M_{tors} → 0

splits and gives the desired result.

Remark. Proposition 10 fails if ℵ_{0} is replaced by a finite cardinal or by an uncountable one. It is of course possible to have finitely generated torsion-free modules which are not free, if A is not a principal ideal domain. In the other direction, an infinite direct product of copies of ℤ is a torsion-free tame ℤ-module of uncountable rank, but it is not free. (See Theorem 19.2 in [5].)

Acknowledgements

I thank D. Goss for encouragement, for posing the questions considered in Section 6, and for suggesting numerous improvements to this paper. I also thank D. Goss, T. Y. Lam, and H. Lenstra for directing me to relevant references, and my advisor K. Ribet for expositional suggestions. Finally, I thank M. Rosen for permission to use his remark on finitely generated torsion modules over Dedekind domains, mentioned after the proof of Theorem 1.

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Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720

E-mail address: poonen@math.berkeley.edu