TORSION IN RANK 1 DRINFELD MODULES AND THE UNIFORM BOUNDEDNESS CONJECTURE

BJORN POONEN

Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld A-module of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an \overline{L}-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over L which have nonzero torsion. For the case A = 𝔽_{q}[T], r = 1, and L = 𝔽_{q}(T), we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the 𝔭-primary part of the torsion for rank 2 Drinfeld 𝔽_{q}[T]-modules over a fixed function field.

1. Conjectures and Theorems

In a 1977 paper, Mazur [13] proved that if E is an elliptic curve over ℚ, its torsion subgroup is one of the following fifteen groups:

ℤ/Nℤ, 1 ≤ N ≤ 10 or N = 12;
ℤ/2ℤ × ℤ/2Nℤ, 1 ≤ N ≤ 4.

In particular, there is a uniform bound for the size of the torsion subgroup of an elliptic curve over ℚ. (Although this in recent years has often been called Ogg’s conjecture, it was essentially formulated by Levi [11] in 1908, and was again formulated by Nagell [15] in 1949, before the appearance of Ogg’s paper [16] in 1971. See [19] for more history.)

Recently, this has been generalized to arbitrary number fields. Merel [14] proved that for every d ≥ 4, the order of the torsion subgroup of an elliptic curve E over a number field K of degree d over ℚ is divisible only by primes less than 2^{d+1}(d!)^{5d/2}. Earlier results of Kamienny and Mazur [10] and Abramovich [1] established bounds for d ≤ 14. Also, Kamienny and Mazur [10] proved that the exponent of each such prime is bounded, with the bound depending only on d. (Earlier work of Manin [12] proved the boundedness of each exponent for fixed K.) Together, these results imply that there is a bound N(d) depending only on d such that the order of the torsion subgroup of an elliptic curve over a number field of degree d over ℚ is at most N(d). This paper considers the analogous questions for Drinfeld modules.

Date: June 19, 1995.

1991 Mathematics Subject Classification. Primary 11G09.

Key words and phrases. Drinfeld module, torsion, uniform boundedness, local height function, signnormalized.

This research was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. Research at MSRI is supported in part by NSF grant DMS-9022140.

For an introduction to the theory of Drinfeld modules see [9] or Drinfeld’s original paper [4]. As usual in this theory, let A denote the ring of regular functions on the affine curve obtained by deleting a closed point “∞” from a nonsingular projective curve X over 𝔽_{q}. Let K be the fraction field of A. In this paper, we will always equip a field extension L of K with the inclusion homomorphism A → L. In this context it makes sense to speak of rank r Drinfeld A-modules over L, for each r ≥ 1. If ϕ is such a Drinfeld module, ^{ϕ}L will denote the additive group of L equipped with the A-module structure coming from ϕ, and (^{ϕ}L)_{tors} will denote its torsion submodule; i.e.,

(^{ϕ}L)_{tors} = { α ∈ L | ϕ_{a}(α) = 0 for some nonzero a ∈ A }.

Finally, #S will denote the cardinality of a set S. It is known (see [3], for example) that if L is a finite extension of K, then #(^{ϕ}L)_{tors} < ∞.

The following two conjectures (a weak and a strong form) would give natural analogues of the recent results on uniform boundedness for elliptic curves.

Conjecture 1. For fixed A, r ≥ 1, and finite extension L of K, there is a uniform bound on #(^{ϕ}L)_{tors} as ϕ ranges over rank r Drinfeld A-modules over L.

Conjecture 2. For fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on #(^{ϕ}L)_{tors} as L ranges over extensions of K of degree less than or equal to d, and ϕ ranges over rank r Drinfeld A-modules over L.

One might ask if there is a uniform bound over an even larger class of Drinfeld modules. It is certainly not reasonable to ask for a uniform bound if d is not fixed, because for any Drinfeld module ϕ over a function field L, the torsion submodule over \overline{L} is infinite. If L is fixed, but r is allowed to vary, then again one cannot hope for a uniform bound on the torsion, because if V is any finite sub-𝔽_{q}-vector space in K = 𝔽_{q}(T), then there is a polynomial of the form

ϕ_{T} (x) = T x + a_{1}x^{q} + a_{2}x^{q^{2}} + ⋯ a_{r}x^{q^{r}} ∈ K[x]

with zero set V, and this defines a Drinfeld 𝔽_{q}[T]-module ϕ over K whose torsion submodule is of size at least #V.

The first main goal of this paper is to prove the following partial result.

Theorem 1. Conjecture 2 holds when r = 1 (for each A and d).

In fact, we will show that for almost all rank 1 Drinfeld modules, there is no nontrivial torsion at all!

Theorem 2. For fixed A and finite extension L of K, there are at most finitely many rank 1 Drinfeld A-modules ϕ over L (up to isomorphism over L) for which (^{ϕ}L)_{tors} ≠ 0.

Theorem 2 fails for rank r ≥ 2. For example, if A = 𝔽_{q}[T], L = K = 𝔽_{q}(T), then each α ∈ L^{∗} gives rise to a rank 2 Drinfeld module

ϕ_{T} (x) = T x − (α + T)x^{q} + αx^{q^{2}}

for which 1 ∈ ^{ϕ}L is torsion. In fact, this gives infinitely many examples with good reduction at a specified place.

For fixed A, there are only finitely many rank 1 Drinfeld modules over L up to isomorphism over the algebraic closure \overline{L}. In fact, their number is the class number of A. But each of these \overline{L}-isomorphism classes may consist of infinitely many distinct classes of Drinfeld modules up to isomorphism over L, and these may have different torsion submodules, as in Theorem 9 below, for example, so these theorems do not follow trivially even if L is fixed. On the other hand, the finiteness of the number of \overline{L}-isomorphism classes implies that Theorems 1 and 2 are corollaries of the following two theorems, which apply to Drinfeld modules of arbitrary rank. A twist of a Drinfeld module ϕ over L′ is a Drinfeld module over L′ which becomes isomorphic to ϕ over the algebraic closure.

Theorem 3. Let ϕ be a Drinfeld A-module over a finite extension L of K. Also fix d ≥ 1. There exists a uniform bound C > 0 (depending on ϕ, A, L, and d) such if ψ is a twist of ϕ over a field extension L′ with [L′ : L] ≤ d, then #(^{ψ}L′)_{tors} ≤ C.

Theorem 4. If ϕ is a Drinfeld module over a finite extension L of K, then at most finitely many of its twists ψ over L have (^{ψ}L)_{tors} ≠ 0.

The elliptic curve analogue of Theorem 4, namely that the torsion subgroups of almost all twists of a fixed elliptic curve over a number field are trivial, is not quite true, but becomes true if we ignore 2-torsion (and also 3-torsion in the case of zero j-invariant).

In fact, we can generalize Theorem 3 further, to the following analogue of a known result for abelian varieties. (To deduce Theorem 3 from the following, let 𝔭 be a prime of K above which lies a prime of good reduction of ϕ, and observe that every twist of ϕ then has potential good reduction at a place above 𝔭.)

Theorem 5. Fix r, d ≥ 1, fix A, and fix a prime 𝔭 of A. There exists C > 0 such that if L is a finite extension of K of degree at most d and ϕ is a rank r Drinfeld A-module over L with potential good reduction at some prime above 𝔭, then #(^{ϕ}L)_{tors} < C.

But, as was remarked after the statement of Theorem 2, we cannot generalize Theorem 4 in the same way.

Sections 2, 3 and 4 give three different approaches to proving Theorem 1. The first two, which use local fields and adelic parallelotopes respectively, also prove Theorem 2 and the more general versions (Theorems 3 and 4), and the first also proves Theorem 5. The third, which uses the explicit description of the action of Galois on the abelian extensions obtained by adjoining torsion of sign-normalized rank 1 Drinfeld modules, proves only Theorem 1.

Each method of proof has its own virtues. (So there is reason for providing several proofs for the same theorems, beyond that of simply desiring to exhibit the various techniques that can be used to attack such problems.) The proofs involving local fields are the quickest, and more importantly are the only ones which give Theorem 5. The proofs involving adelic parallelotopes require only computations within the field over which the Drinfeld modules of interest are defined, and this makes them especially suitable for working out specific examples, as we will do in Section 5. The proof involving class field theory gives the best bounds (just slightly worse than linear in the degree of the field, as opposed to exponentially large). In fact, the bounds it gives will be proven to be best possible, up to a constant factor.

Section 6 explains why these three methods fail to verify the conjectures for r ≥ 2, and proves the following partial result for rank 2, which is analogous to Manin’s local result for elliptic curves over number field mentioned above.

Theorem 6. Let A = 𝔽_{q}[T] and K = 𝔽_{q}(T). Fix a prime 𝔭 of A and a finite extension L of K. Then there is a uniform bound on the size of the 𝔭-primary part of the torsion of rank 2 Drinfeld A-modules over L.

2. Proofs using local fields

As mentioned in the introduction, whenever we have a Drinfeld module over an extension of K, it is understood that the ring homomorphism A → L is the inclusion map.

Proposition 1. Let L be a finite extension of the completion K_{𝔭} of K at a place 𝔭 other than ∞, and let ϕ be a Drinfeld module over L. Then (^{ϕ}L)_{tors} is finite.

Proof. If we fix a non-constant T ∈ A, then we may consider ϕ as a Drinfeld 𝔽_{q}[T]-module over L, and the torsion submodule will be the same subset of L, since the torsion submodule can be characterized as the set of elements which have finite orbit under iteration of ϕ_{T}. Thus without loss of generality assume A = 𝔽_{q}[T].

Let v be the discrete valuation on L. If v(x) is sufficiently negative, then v(ϕ_{T} (x)) is more negative, and is determined by the leading term of ϕ_{T}. From this, if g ∈ A = 𝔽_{q}[T], v(ϕ_{g}(x)) will be determined by the leading term in g. In particular (still assuming v(x) is sufficiently negative), ϕ_{g}(x) will not be zero. Hence torsion elements are bounded below in valuation, and their closure is compact. On the other hand, there are no nonzero torsion elements in a neighborhood of zero, because the action of ϕ_{T} near zero is locally analytically conjugate via a formal A-module isomorphism to multiplication-by-T, and |T | ≤ 1 with respect to the absolute value on L, so in a sufficiently small ball around 0, the entire Drinfeld A-action is analytically conjugate to the standard A-action. Thus the group (^{ϕ}L)_{tors} is discrete as well as compact, so it is finite.

Remark . The lemma could sometimes fail if L were allowed to be K_{∞} or a finite extension thereof, as can be seen from the analytic uniformization theory. (The same thing happens for elliptic curves: an elliptic curve over a p-adic field has finite torsion, whereas an elliptic curve over ℝ or ℂ has infinite torsion.)

Theorem 5 follows immediately from the following quantitative version of Proposition 1 for Drinfeld modules of potential good reduction.

Theorem 7. Fix r, d ≥ 1, fix A, and fix a prime 𝔭 of A. There exists C > 0 such that if L is a finite extension of K_{𝔭} of degree at most d and ϕ is a rank r Drinfeld A-module over L with potential good reduction, then #(^{ϕ}L)_{tors} < C.

Proof. Pick T ∈ 𝔭. Any Drinfeld A-module can be considered as a Drinfeld 𝔽_{q}[T]-module (with different but still bounded r and d), so we may assume A = 𝔽_{q}[T], and K_{𝔭} = 𝔽_{q}((T)). This time we equip any finite extension L of K_{𝔭} with the valuation v normalized by v(T) = 1. If ϕ is a rank r Drinfeld 𝔽_{q}[T]-module over L with potential good reduction, then over the field obtained by adjoining the (q^{r} − 1)-th root of the leading coefficient of ϕ_{T} to L, it is isomorphic to a Drinfeld module with good reduction. Hence again replacing d by something larger, but bounded, we may assume ϕ has good reduction over L. In other words,

ϕ_{T} (x) = T x + (higher powers of x) ∈ O[x],

where O is the ring of integers of L. All torsion of ϕ over L is contained in O. If x is in the ideal T^{2}O, then v(ϕ_{T} (x)) = v(x)+1, and then by induction we see that v(ϕ_{T^{k}}(x)) = v(x)+k, which implies that the orbit of x under ϕ_{T} is infinite, and hence x cannot be torsion. Thus (^{ϕ}L)_{tors} injects (as a group) into O/T^{2}O, which is of size q^{2[L:K_{𝔭}]} ≤ q^{2d}.

First proof of Theorem 2. Again we may assume A = 𝔽_{q}[T]. Write

ϕ_{T} (x) = T x + a_{1}x^{q} + a_{2}x^{q^{2}} + ⋯ a_{r}x^{q^{r}}.

Every twist ψ of ϕ over L is of the form

ψ_{T} (x) = T x + u^{q−1}a_{1}x^{q} + u^{q^{2}−1}a_{2}x^{q^{2}} + ⋯ u^{q^{r}−1}a_{r}x^{q^{r}},

for some u ∈ \overline{L}^{∗}. In fact, the requirement that ψ_{T} have coefficients in L forces u^{n} ∈ L, where n is the GCD of q^{i} − 1 over all i for which a_{i} ≠ 0. It is easy to see that n also equals # Aut(ϕ).

Let L_{v} be the completion of L at some finite place, and let F be the field extension obtained by adjoining to L_{v} all n-th roots of all elements. Since n is prime to char L_{v}, Kummer theory implies that F is a finite extension of L_{v}. Every twist ψ of ϕ is isomorphic to ϕ over F, so we obtain an isomorphism

ψF ⟶^{u^{−1}} ^{ϕ}F.

If ψ and ψ′ are two such Drinfeld modules, then the images of ^{ψ}L and ^{ψ′}L intersect nontrivially only if u ∈ L, in which case ψ and ψ′ are isomorphic over L. Hence the union of (^{ψ}L)_{tors} \ {0}, with ψ ranging over the distinct twists of ϕ over L, injects as a set into (^{ϕ}F)_{tors}, which is finite by Proposition 1. Hence there can be only finitely many such twists ψ for which (^{ψ}L)_{tors} is nontrivial.

3. Proofs using adelic parallelotopes

The torsion submodule of a Drinfeld module over a global field can be characterized as the set of elements where the (global) canonical height is zero, or equivalently where each canonical local height is zero. (See [3] for the global, [17] for the local.) The latter condition can be satisfied at a given place v only when the corresponding absolute value is small, and thus we can construct an adelic parallelotope in which all torsion lies. Making these estimates uniformly over varying Drinfeld modules leads to proofs of our theorems.

For each place 𝔭 of a global function field L, let ∥ ∥_{𝔭} denote the associated absolute value, normalized so that for each x in the valuation ring O_{𝔭} of 𝔭, ∥x∥ = #(O_{𝔭}/x). The first lemma below bounds the torsion of a Drinfeld module within a certain adelic parallelotope, and the next bounds the number of field elements within such a parallelotope.

Lemma 1. Let ϕ be a Drinfeld A-module over L. Then for each place 𝔭 of L, there exists c_{𝔭} > 0 such that for any extension | | of ∥ ∥_{𝔭} to a finite extension E of L, every α ∈ (^{ϕ}E)_{tors} satisfies |α| ≤ c_{𝔭}. Moreover one can take c_{𝔭} = 1 for all but finitely many 𝔭.

Proof. The case E = L is essentially a restatement of (4) in Proposition 4 and (3) in Proposition 1 of [17], together with the observation (following from (3) in Proposition 4 of [17]) that if some canonical local height function is nonzero at α, then α is not in the torsion submodule. It is clear from the proofs there that the constant c_{𝔭} depends only on the absolute values of the coefficients of the polynomials defining the Drinfeld module, so a single constant can be chosen to work for all E. It is important here that | | is literally an extension of ∥ ∥_{𝔭} rather than the corresponding normalized absolute value, which is a power of | |.

Lemma 2. For each place 𝔭 of a global function field L, let x_{𝔭} be a positive real number, such that x_{𝔭} = 1 for all but finitely many 𝔭. Fix one such place 𝔮, and let k_{𝔮} denote its residue field. Let M be the number of α ∈ L such that ∥α∥_{𝔭} ≤ x_{𝔭} for all 𝔭. Then either M = 1 (i.e., the only such α is 0), or

M ≤ #k_{𝔮} ∏_{𝔭}x_{𝔭}.

Proof. This is a special case of Lemma 8 in [2]. The values of the constants in that lemma for this special case come out of its proof and case 3 of the proof of Lemma 7 there.

Proof of Theorem 3. Just as in the proofs in Section 2, we may assume that A = 𝔽_{q}[T], at the expense of increasing the rank. Write

ϕ_{T} (x) = T x + a_{1}x^{q} + a_{2}x^{q^{2}} + ⋯ a_{r}x^{q^{r}}.

Every twist ψ over L′ isomorphic to ϕ over \overline{L} is of the form

ψ_{T} (x) = T x + u^{q−1}a_{1}x^{q} + u^{q^{2}−1}a_{2}x^{q^{2}} + ⋯ u^{q^{r}−1}a_{r}x^{q^{r}},

for some u ∈ \overline{L}^{∗}. The torsion submodule of ψ over L′ consists of those α ∈ L′ for which uα is a torsion element of ϕ. Let E = L′(u).

Suppose uα is a torsion element of ϕ. For each place 𝔭 of L, pick a constant c_{𝔭} as in Lemma 1 for ϕ. Let 𝔓 be a place of E above the places 𝔭′ of L′ and 𝔭 of L, and let n(𝔓/𝔭) denote local degree. Then ∥ ∥_{𝔓}^{n(𝔓/𝔭)^{−1}} is an absolute value on E extending ∥ ∥_{𝔭}. So by Lemma 1,

∥uα∥_{𝔓}^{n(𝔓/𝔭)^{−1}} ≤ c_{𝔭},
(1)  ∥α∥_{𝔓} ≤ c_{𝔭}^{n(𝔓/𝔭′)n(𝔭′/𝔭)}∥u∥_{𝔓}^{−1},

Now fix 𝔭′. If one takes the product of (1) over all places 𝔓 of E above 𝔭′, one obtains

∥α∥_{𝔭′}^{[E:L′]} ≤ c_{𝔭}^{[E:L′]n(𝔭′/𝔭)} ∏_{𝔓|𝔭′}∥u∥_{𝔓}^{−1},
(2)  ∥α∥_{𝔭′} ≤ c_{𝔭}^{n(𝔭′/𝔭)} ∏_{𝔓|𝔭′}∥u∥_{𝔓}^{−1/[E:L′]}.

Fix any prime 𝔮 of L, and let 𝔮′ be an extension to L′. By Lemma 2, the number of α ∈ L′ satisfying (2) for all places 𝔭′ of L′ is at most

#k_{𝔮′} ∏_{𝔭′} (c_{𝔭}^{n(𝔭′/𝔭)} ∏_{𝔓|𝔭′}∥u∥_{𝔓}^{−1/[E:L′]})

≤ (#k_{𝔮})^{[L′:L]} (∏_{𝔭}c_{𝔭}^{[L′:L]}) (∏_{𝔓}∥u∥_{𝔓}^{−1/[E:L′]})

= (#k_{𝔮} ∏_{𝔭}c_{𝔭})^{[L′:L]} ,

by the product formula in E applied to u. Thus the size of the torsion submodule of ψ over L′ is at most C^{[L′:L]} ≤ C^{d}, where

C = #k_{𝔮} ∏_{𝔭}c_{𝔭}.

Second proof of Theorem 2. Again we may assume A = 𝔽_{q}[T]. Write

ϕ_{T} (x) = T x + a_{1}x^{q} + a_{2}x^{q^{2}} + ⋯ a_{r}x^{q^{r}}.

Every twist ψ of ϕ over L is of the form

ψ_{T} (x) = T x + f^{(q−1)/n}a_{1}x^{q} + f^{(q^{2}−1)/n}a_{2}x^{q^{2}} + ⋯ + f^{(q^{r}−1)/n}a_{r}x^{q^{r}}, (3)

where n is as in the first proof of Theorem 3 and f ∈ L^{∗}, and the isomorphism type of ψ depends only on the image of f in L^{∗}/L^{∗n}.

We claim that there exists a finite set of places S of L and integers m_{v} for each v ∈ S such that any g ∈ L^{∗} can be multiplied by an n-power to obtain f satisfying

1. v(f) ≤ m_{v} if v ∈ S.

2. 0 ≤ v(f) ≤ n − 1 if v ∉ S.

Let D denote the group of degree zero divisors on the curve of which L is the function field, and let P denote the subgroup of principal divisors. Then the class group D/P of L is finite, and multiplication by n maps it surjectively onto nD/nP, so nD/nP is finite as well. Let D_{1}, …, D_{s} be representatives for the cosets of nP in nD. Let S be the set of places of L which occur in D_{1}, …, D_{n}, and let v_{0} be one of these places. (If S is empty, enlarge it so it contains at least one place.) Given g, with principal divisor (g) = ∑ α_{v}v, let D be the degree zero divisor D = ∑ α′_{v}v where α′_{v} = n⌊α_{v}/n⌋ for v′ ≠ v_{0}, and α′_{v_{0}} is set to make the total degree 0. Then by definition of the D_{i}, we have −D = D_{i} + n(u) for some i and some u ∈ L^{∗}. So f =^{def} gu^{n} satisfies

(f) = (g) + n(u) = (g) − D′ − D_{i}.

The coefficient β_{v} of v in (f) is α_{v} mod n if v ∉ S. If v ∈ S but v ≠ v_{0}, then β_{v} is bounded above and below, where the bound depends on the D_{i} but not on g. Since ∑_{v} β_{v} = 0, and we have lower bounds on each β_{v} for v ≠ v_{0} which are zero for all but finitely many v, we get also an upper bound on β_{v_{0}}, as desired.

From now on, we only consider twists ψ of the form (3) with f satisfying the two conditions of the previous paragraph. We claim there is an adelic parallelotope of L (of finite volume) which contains all torsion elements of all such ψ. Let T be the (finite) set of places of L which occur in the factorizations of the nonzero a_{i}.

First suppose v ∉ S ∪ T. Then v(f) ≤ n − 1 by condition 1 above. If v(x) ≤ −1, then

v(f^{(q^{r}−1)/n}a_{r}x^{q^{r}}) − v(f^{(q^{i}−1)/n}a_{i}x^{q^{i}}) = (q^{r} − q^{i})/n · v(f) + (q^{r} − q^{i})v(x)
≤ (q^{r} − q^{i})(n − 1)/n − (q^{r} − q^{i})
< 0,

so the term f^{(q^{r}−1)/n}a_{r}x^{q^{r}} is larger v-adically than all the other terms in (3). In particular it is larger than the term T x, so v(ψ_{T} (x)) < v(x). Hence x tends to infinity v-adically under iteration of ψ_{T}, and so cannot be a torsion element. In other words, the torsion of ψ over L lies within the set where v(x) ≥ 0.

Now suppose v ∈ S ∪ T. Since v(f) ≤ m_{v}, we still have

v(f^{(q^{r}−1)/n}a_{r}x^{q^{r}}) − v(f^{(q^{i}−1)/n}a_{i}x^{q^{i}}) < 0

provided that v(x) < c_{v} where c_{v} is sufficiently negative. Then the same argument in the previous paragraph shows that the torsion of ψ lies within the set where v(x) ≥ c_{v}.

Thus we have shown that all torsion of ψ over L lies within an adelic parallelotope whose bounds are independent of f, hence independent of ψ. By Lemma 2, there are only finitely many elements α_{1}, …, α_{M} within this adelic parallelotope. If α_{i} is a nonzero torsion element for ψ for a particular f, then ψ_{T} (α_{i}) is also torsion, so ψ_{T} (α_{i}) = α_{j} for some j. Rewriting this using (3) shows that f satisfies a algebraic equation, so there are finitely many possibilities for f, for each α_{i} and α_{j}. The (finite) union of all these over i and j lists every f for which the twist ψ might have a nonzero torsion element.

4. Proofs using explicit class field theory

The bounds on the size of the torsion submodule of rank 1 Drinfeld modules given by the proofs of the last two sections grow exponentially in the degree d. In this section we will give a third proof of Theorem 1, and this one will give a much better bound (just slightly worse than linear in d). In fact, the bound we obtain here will be shown to be best possible, up to a constant factor.

The basic tool we use in this section is the explicit description of the action of Galois on the torsion of a sign-normalized Drinfeld module. This lets us determine which torsion elements belong to a particular global field. The main drawback of these methods is that they apply only to rank 1 Drinfeld modules, and hence do not also give a proof of Theorem 3 on twists of Drinfeld modules of higher rank. The quantitative version of Theorem 1 we prove is the following:

Theorem 8. For fixed A, there is a constant C > 0 such that if ϕ is a rank 1 Drinfeld A-module over a global field L with [L : K] = d, then

#(^{ϕ}L)_{tors} ≤ C · d log log d.

For each A, this bound is best possible up to a constant factor.

In the proof, we will need bounds on a characteristic p analogue of the Euler phi-function. If 𝔪 is an ideal of A, we let |𝔪| = #A/𝔪, deg 𝔪 = dim_{𝔽_{q}} A/𝔪 = log_{q} |𝔪|, and φ_{A}(𝔪) = #(A/𝔪)^{∗}. Just as in the classical case, we have

φ_{A}(𝔪) = |𝔪| · ∏_{𝔭|𝔪}(1 − 1/|𝔭|). (4)

Lemma 3. For each A,

lim inf_{|𝔪|→∞} \frac{φ_{A}(𝔪) log log |𝔪|}{|𝔪|}

is a (finite) positive real number.

Proof. First we estimate the number of primes of A of given degree. Recall that A is the ring of regular functions on a nonsingular projective curve X minus a closed point ∞. Degree m primes of A correspond to Gal(\overline{𝔽}_{q}/𝔽_{q})-conjugacy classes of points of X (other than ∞) defined over 𝔽_{q^{m}} but not over any smaller field. By the Riemann hypothesis for curves over finite fields, #X(𝔽_{q^{m}}) = q^{m} + O(q^{m/2}) as m goes to infinity. (The constant implied by the big-O depends on X.) Of the points defined over 𝔽_{q^{m}}, the number defined over smaller fields is at most

∑_{s|m,s≠m} #X(𝔽_{q^{s}}) ≤ ∑_{s=1}^{m/2} [q^{s} + O(q^{s/2})] = O(q^{m/2})

so there remain q^{m} +O(q^{m/2}) whose field of definition is 𝔽_{q^{m}}. Hence the number of primes of A of degree m is q^{m}/m + O(q^{m/2}/m). (It does not matter that we ignored the single prime ∞.)

Consider the product 𝔪 of the primes of degree up to n. As is standard, Θ(f(n)) denotes a function which is bounded above and below by positive multiples of f(n). We have

deg 𝔪 = ∑_{m=1}^{n}(q^{m}/m + O(q^{m/2}/m))m = Θ(q^{n})

(5) log log |𝔪| = log log q^{Θ(q^{n})} = n log q + O(1)

and

∑_{deg 𝔭≤n}log(1 − 1/|𝔭|) = ∑_{m=1}^{n}(q^{m}/m + O(q^{m/2}/m)) log(1 − q^{−m})

= ∑_{m=1}^{n}(q^{m}/m + O(q^{m/2}/m))(−q^{−m} + O(q^{−2m}))

= ∑_{m=1}^{n}(−1/m + O(q^{−m/2}))

= − log n + O(1)

(6)  ∏_{deg 𝔭≤n}(1 − 1/|𝔭|) = Θ(1/n).

(The estimate (6) is an analogue of Mertens’ Theorem which was proved independently by Rosen [18], who also computes the constant implied by the Θ notation.) By (4), (5) and (6),

\frac{φ_{A}(𝔪) log log |𝔪|}{|𝔪|} = (n log q + O(1))Θ(1/n) = Θ(1),

which proves that the lim inf is finite.

On the other hand, for any ideal 𝔪 of degree n, the number k of primes 𝔭_{1}, …, 𝔭_{k} dividing 𝔪 having degree at least log_{q} n is at most n/ log_{q} n, so by (4),

φ_{A}(𝔪)/|𝔪| = ∏_{i=1}^{k}(1 − 1/|𝔭_{i}|) ∏_{𝔭|𝔪,deg 𝔭<log_{q} n}(1 − 1/|𝔭|)

≥ (1 − 1/q^{log_{q} n})^{n/ log_{q} n} ∏_{deg 𝔭<log_{q} n}(1 − 1/|𝔭|)

≥ (1 − 1/ log_{q} n)Θ(1/⌊log_{q} n⌋) (by (6))

= Θ(1/ log n)

= Θ(1/ log log_{q} |𝔪|) = Θ(1/ log log |𝔪|),

which proves that the lim inf is positive.

Corollary 1. For each A, there is a constant c > 0 such that

|𝔪| ≤ c φ_{A}(𝔪) log log φ_{A}(𝔪)

for every ideal 𝔪 of A.

Proof of Theorem 8. Fix a sign-function, and let H_{A}^{+} be the corresponding normalizing field for rank 1 Drinfeld A-modules, which is the minimal field of definition for each sign-normalized Drinfeld A-module. (See [9] for definitions.) Suppose ϕ is a rank 1 Drinfeld A-module over L, and [L : K] = d. By Theorem 12.3 in [9], ϕ is isomorphic to some sign-normalized Drinfeld module ψ, which is defined over H_{A}^{+}. Let E be a compositum of L and H_{A}^{+} in some algebraic closure of K. Then ϕ and ψ are twists over E, and they become isomorphic over a field extension F obtained by adjoining a single n-th root, where n (as in the proofs of Section 2) equals # Aut(ϕ) = q − 1, by Corollary 5.14 in [9]. In particular,

(^{ϕ}L)_{tors} ⊂ (^{ϕ}F)_{tors} ≅ (^{ψ}F)_{tors}.

Since ψ is of rank 1, (^{ψ}F)_{tors} is isomorphic as an A-module to A/𝔪 for some ideal 𝔪 of A. Let K_{𝔪} be the narrow ray class field, which is the field generated over H_{A}^{+} by the 𝔪-torsion of ψ. Section 16 of [9] gives us an isomorphism from Gal(K_{𝔪}/H_{A}^{+}) to (A/𝔪)^{∗}, so

φ_{A}(𝔪) = # Gal(K_{𝔪}/H_{A}^{+})
= [K_{𝔪} : H_{A}^{+}]
≤ [F : H_{A}^{+}]
≤ n[E : H_{A}^{+}] (by definition of F)
= n[LH_{A}^{+} : KH_{A}^{+}]
≤ nd.

Finally

#(^{ϕ}L)_{tors} ≤ #(^{ψ}F)_{tors}
= |𝔪|
≤ cφ_{A}(𝔪) log log φ_{A}(𝔪) (by Corollary 1)
≤ c(nd) log log(nd)
≤ C · d log log d,

for some constant C depending only on A.

This is best possible, because for the sequence of 𝔪’s which make the lim inf in Lemma 3 small, if ϕ is a sign-normalized Drinfeld A-module over the narrow ray class field K_{𝔪}, then

d = [K_{𝔪} : K] = [K_{𝔪} : H_{A}^{+}][H_{A}^{+} : K] = O(φ_{A}(𝔪)) = O(|𝔪|/ log log |𝔪|)

while #(^{ϕ}L)_{tors} ≥ |𝔪|.

5. Examples

Theorem 2 tells us in particular that over K = 𝔽_{q}(T), there are only finitely many rank 1 Drinfeld 𝔽_{q}[T]-modules that have nonzero torsion. Here we list them explicitly.

Theorem 9. Let A = 𝔽_{q}[T] and K = 𝔽_{q}(T). Let ϕ be a rank 1 Drinfeld A-module over K.

Case 1: q = 2

In this case ϕ is isomorphic to the Carlitz module ϕ_{T} (x) = T x + x^{2} over K. For the Carlitz module, (^{ϕ}K)_{tors} = 𝔽_{2} + 𝔽_{2} · T, and the torsion is isomorphic to 𝔽_{2}[T]/(T^{2} + T) as an 𝔽_{2}[T]-module.

Case 2: q > 2

If ϕ_{T} (x) = T x − (T + c)x^{q} for some c ∈ 𝔽_{q}, then the torsion submodule (^{ϕ}K)_{tors} equals 𝔽_{q} ⊂ ^{ϕ}K and is isomorphic to 𝔽_{q}[T]/(T + c) as an 𝔽_{q}[T]-module. If ϕ is not isomorphic over K to such a Drinfeld module, then (^{ϕ}K)_{tors} = 0.

We will specialize the second proof of Theorem 4, since it is the only one that does not require that we work within field extensions larger than K. The fact that K has class number 1 will simplify matters greatly.

Proof. Let v_{g} denote the ℤ-valued discrete valuation on K associated with an irreducible element g of 𝔽_{q}[T]. The only nontrivial valuation not of this form is the one at ∞, the degree function deg.

We have

ϕ_{T} (x) = T x − fx^{q}

for some f ∈ 𝔽_{q}(T). Since ϕ is isomorphic over K to any Drinfeld module ψ with

ψ_{T} (x) = T x − u^{q−1}fx^{q}

for u ∈ K^{∗}, we may assume without loss of generality that f ∈ 𝔽_{q}[T]. Also, since 𝔽_{q}[T] is a UFD, we may assume that v_{g}(f) < q − 1 for each irreducible f ∈ 𝔽_{q}[T]. (When q = 2, this means that f = 1 and ϕ is the Carlitz module.) The set S ∪ T in the second proof of Theorem 2 for us is just {∞}. Then that proof gives v_{g}(α) ≥ 0 for all irreducible g in 𝔽_{q}[T]. In other words, (^{ϕ}K)_{tors} ⊂ 𝔽_{q}[T].

Suppose α ⊂ 𝔽_{q}[T] is of degree at least 1 if q ≠ 2, and at least 2 if q = 2. Then deg α^{q−1} ≥ 2, so

deg(fα^{q}) > deg(T α)
deg(ϕ_{T} (α)) = deg(fα^{q}) > deg α
deg(ϕ_{T^{n}}(α)) → ∞

as n goes to ∞. This implies that α is not a torsion element, because a torsion element has finite orbit under iteration ϕ_{T}. Thus (^{ϕ}K)_{tors} ⊂ 𝔽_{q} if q ≠ 2, and (^{ϕ}K)_{tors} ⊂ 𝔽_{2} + 𝔽_{2} · T if q = 2.

For q = 2 (and f = 1),

ϕ_{T +1}(1) = T
ϕ_{T} (T) = 0

so (^{ϕ}K)_{tors} = 𝔽_{2} + 𝔽_{2} · T and it is a cyclic module generated by 1 and isomorphic to 𝔽_{2}[T]/(T^{2} + T).

For q > 2, (^{ϕ}K)_{tors} ⊂ 𝔽_{q}. If f = T + c with c ∈ 𝔽_{q}, then ϕ_{T +c}(1) = 0, so it is clear that (^{ϕ}K)_{tors} = 𝔽_{q}, and that it is isomorphic to 𝔽_{q}[T]/(T + c) as an 𝔽_{q}[T]-module. Otherwise ϕ_{T} (1) = T − f ∉ 𝔽_{q}, so 1 ∉ (^{ϕ}K)_{tors}, and (^{ϕ}K)_{tors} = 0.

6. Uniform bounds for Drinfeld modules of higher rank

In this section we will first explain why the methods of the previous three section fail to prove Conjectures 1 or 2 in general for Drinfeld modules of rank r ≥ 2, and then prove Theorem 6, which is the only result beyond Theorem 5 that we have in this situation. If we work over a local field (the completion of the global function field at a prime not above ∞), we still have finiteness of the torsion submodule for each individual Drinfeld module by Proposition 1, but on the other hand we will be able to exploit the Tate uniformization Drinfeld modules in the next proposition to construct examples where the torsion submodule over the local field can be arbitrarily large. Let H_{A} denote the Hilbert class field of A, which is the maximal unramified abelian extension of K in which ∞ splits completely. (See Section 15 of [9].)

Proposition 2. Let L be a finite extension of the completion of H_{A} at a place not above ∞. Then for each r ≥ 2 and nonzero a ∈ A, there exists a rank r Drinfeld module ϕ over L with a torsion element of order a.

Proof. There exists a rank 1 Drinfeld A-module ψ over L, since the minimal field of definition for any rank 1 Drinfeld A-module is H_{A}. (See [9].) By replacing ψ with an isomorphic Drinfeld module, we may assume the coefficients of ψ_{a} for all a lie in the valuation ring O of L. Then O and L both inherit A-module structures from ϕ, and the quotient ^{ϕ}L/^{ϕ}O as an A-module is isomorphic to the direct sum of a free module of rank ℵ_{0} with a finite torsion module, by Theorem 2 in [17]. Let λ_{1}, λ_{2}, …, λ_{r−1} be elements of^{ϕ}L whose images in^{ϕ}L/^{ϕ}O are A-independent. The A-module Λ generated by the λ_{i} is discrete since its intersection with ϕO is trivial. Let ϕ_{a}(Λ) denote the submodule generated by ϕ_{a}(λ_{1}), ϕ_{a}(λ_{2}), …, ϕ_{a}(λ_{r−1}). Then the quotient of ψ by ϕ_{a}(Λ) can be given the structure of a rank r Drinfeld module over L, as in Section 7 of [4] on Tate uniformization, and the image of λ_{1} in this quotient has order a.

The adelic parallelotope method fails when r ≥ 2, simply because the volume of the parallelotope needed to contain the torsion can be arbitrarily large. For example, suppose A = 𝔽_{q}[T], L = K = 𝔽_{q}(T), V is the sub-𝔽_{q}-vector space of K with basis λ_{1}, …, λ_{r}, and ϕ is the rank r Drinfeld A-module with ker ϕ_{T} = V. Then it is not hard to show that the smallest parallelotope of the adeles 𝔸_{K} containing the T -torsion submodule V has volume q^{h((λ_{1}:···:λ_{r}))}, where h(P) denotes the logarithmic Weil height of a point P ∈ ℙ^{r−1}(K), and this height can be arbitrarily large.

We cannot apply the method of Section 4 to obtain uniform bounds for torsion of Drinfeld modules of rank r ≥ 2 until we have more information about the Galois representations arising from such Drinfeld modules. It is perhaps reasonable to expect that the index of the image of the representation is finite for rank r Drinfeld modules ϕ with End ϕ = A, but to use this to obtain uniform boundedness of torsion, it would probably be necessary to bound the index uniformly over Drinfeld modules of a given rank. Not much is known regarding uniform boundedness of the index in the classical case either, for the image of Galois for non-CM elliptic curves.

Let us now give the proof of Theorem 6. Manin’s proof of the number field analogue required an application of the Mordell conjecture, which had not been proved at the time, but he was able to verify the conjecture in the cases he needed. Similarly, we will need the Mordell conjecture over characteristic p function fields, which was proved by Samuel [21]. I thank Ernst-Ulrich Gekeler for explaining that the necessary hypotheses for the application of the Mordell conjecture (genus at least 2, and non-isotriviality) were essentially known.

Proof of Theorem 6. By Corollary 2.20 in [7], if n ≥ 4, the genus of the Drinfeld modular curve X_{0}(𝔭^{n}) exceeds 2, and then the same will hold for the covering X_{1}(𝔭^{n}). In order to apply the characteristic p Mordell conjecture, we also need to know that X_{1}(𝔭^{n}) is nonisotrivial; i.e. that it cannot be defined over a finite field, even after base change. Let J be the Jacobian of X_{1}(𝔭^{n}) for some n ≥ 4. By [8], J has multiplicative reduction at ∞. Then the criterion of Ogg-Néron-Shafarevich implies that J cannot be defined over a finite field, and so neither can X_{1}(𝔭^{n}).

By the Mordell conjecture for characteristic p function fields (Theorem 4 in [21]), X_{1}(𝔭^{n}) has only finitely many L-rational points. In particular, there are only finitely many j ∈ L for which there might exist a rank 2 Drinfeld module (of that j-invariant) with a 𝔭^{n}-torsion point. Hence it will suffice to prove that for fixed j ∈ L, there is a uniform bound for the 𝔭-primary torsion of Drinfeld modules of that j-invariant. All rank 2 Drinfeld modules of a given j-invariant are twists of one another, so their torsion is uniformly bounded by Theorem 3.

Theorem 6 could probably be generalized to general A. This would involve calculating the genus of the Drinfeld modular curves for these A, or at least verifying that the genus is eventually at least 2. Such a calculation can be done by starting with the genus formula for X(1) (VI 5.8 in [5]), and applying the Hurwitz formula, using the computation of the contributions of the ramified points in [6].

Finally, it should be mentioned that Tamagawa [20] has developed a theory for the rank 2 case over A = 𝔽_{q}[T] analogous to Mazur’s theory of Eisenstein ideals for modular curves over ℚ. In particular, he has succeeded in defining the Eisenstein quotient of the Drinfeld modular curves X_{0}(𝔫) over K, and has proved the finiteness of the Mordell-Weil group and the primeto-q part of the Shafarevitch-Tate group for the “minus part” of this Eisenstein quotient. It is not yet clear whether this will lead to information on rational points on the Drinfeld modular curves themselves, which is what is needed to verify the uniform boundedness conjecture for rank 2 Drinfeld 𝔽_{q}[T]-modules. Perhaps it will be necessary to construct analogues of the “winding quotient,” which is used in [14] for the classical case.

Acknowledgements

I thank Dan Abramovich for suggesting that I try to prove Theorem 5. I thank J. Felipe Voloch for correcting my history, and for helping me find references for the characteristic p Mordell conjecture. Finally, thanks go to Ernst-Ulrich Gekeler for explaining the other results needed in the proof of Theorem 6.

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[19] Schappacher, N. and Schoof, R., Beppo Levi and the arithmetic of elliptic curves, preprint, 1995.

[20] Tamagawa, A. The Eisenstein quotient of the Jacobian variety of a Drinfeld modular curve, to appear in Publ. RIMS. Kyoto Univ.

[21] Samuel, P., Compléments à un article de Hans Grauert sur la conjecture de Mordell, I.H.E.S. Publ. Math. 29 (1966), 55–62.

Mathematical Sciences Research Institute, Berkeley, CA 94720-5070, USA

  E-mail address: poonen@msri.org