A WEIL-BARSOTTI FORMULA FOR DRINFELD MODULES

MATTHEW A. PAPANIKOLAS AND NIRANJAN RAMACHANDRAN

Abstract. We study the group of extensions in the category of Drinfeld modules and Anderson’s t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.

1. Introduction and statement of results

In this paper, we investigate extensions of Drinfeld modules using the well-known analogy between abelian varieties and Drinfeld modules. Associated with an abelian variety A are two other group varieties: (i) the dual abelian variety A^{∨}, and (ii) the universal vectorial (or additive) extension A^{♮}.

The Weil-Barsotti formula identifies Ext^{1}(A, 𝔾_{m}) in the category of group schemes with A^{∨} [14]. Likewise, Ext^{1}(A^{∨}, 𝔾_{m}) ≅ A. Theorem 1.1 below provides a Drinfeld module analogue of this result.

There is also an important relationship between the de Rham cohomology (in characteristic zero) of A and the variety A^{♮} [13]. We study generalizations of a de Rham theory for Drinfeld modules based on additive extensions, which has been developed by G. Anderson, P. Deligne, E.-U. Gekeler, and J. Yu [11].

Moreover, Anderson has remarked that Deligne’s 1-motives [9, §10.1] provide a fruitful analogue for Drinfeld modules. Drinfeld modules (or t-modules) include analogues of 𝔾_{m}, abelian varieties, and semi-abelian varieties, all of which are examples of 1-motives. In [9], Deligne defines duality and de Rham realizations of 1-motives using extensions by 𝔾_{m} and 𝔾_{a}, thereby generalizing the already existing theory for abelian varieties. We pursue these ideas in the context of general extensions of Drinfeld modules.

Notation. Let K be a perfect field of characteristic p > 0, and let 𝔽_{q}[t] be the polynomial ring in one variable over the finite field 𝔽_{q} where q = p^{m}. Fix an 𝔽_{q}-linear homomorphism ι: 𝔽_{q}[t] → K with θ := ι(t). Throughout, all Drinfeld modules and tmodules are defined with respect to the map ι, and in particular all Drinfeld modules are 𝔽_{q}[t]-modules.

Let C denote the Carlitz module, C: 𝔽_{q}[t] → K{τ}, defined by C(t) = θ+τ, where K{τ} is the ring of twisted polynomials in K such that for x ∈ K, τx = x^{q}τ.

We take Ext^{1}(·, ·) to be the bifunctor Ext^{1} from the additive category of t-modules to the category of abelian groups. In §2 we see that, for two t-modules E and F, those extensions which induce trivial t-module extensions of their respective tangent spaces comprise a canonical subgroup Ext_{0}^{1}(E, F) ⊂ Ext^{1}(E, F). For a t-module E, we let E^{∨} := Ext_{0}^{1}(E, C).

Date: July 20, 2001.

1991 Mathematics Subject Classification. 11G09.

Key words and phrases. Drinfeld modules, t-modules, extensions.

Theorem 1.1. Let E be a Drinfeld module of rank r ≥ 2.

(a) The group Ext^{1}(E, C) is naturally a t-module of dimension r and sits in an exact sequence of t-modules

0 → E^{∨} → Ext^{1}(E, C) → 𝔾_{a} → 0.

Furthermore, E^{∨} is the Cartier-Taguchi dual t-module associated to E [19], and in particular, E^{∨} is isomorphic to the (r − 1)-st exterior power E^{∧(r−1)} of E.

(b) The group Ext^{1}(E^{∨}, C) is also naturally a t-module of dimension r and sits in an exact sequence

0 → E → Ext^{1}(E^{∨}, C) → 𝔾_{a}^{r−1} → 0.

Moreover, (E^{∨})^{∨} ≅ E.

Because of some interesting similarities with the classical result, we are tempted to call this result a Weil-Barsotti formula for Drinfeld modules. For Drinfeld modules of rank 2, this result has been proved by S. S. Woo [20]. Furthermore, Y. Taguchi shows that one has a Weil pairing on the torsion points of E and E^{∨} [19].

Theorem 1.1 requires us to work outside the category of Drinfeld modules, and one may ask for general t-modules E and F over K whether Ext^{1}(E, F) has the structure of a t-module. In this vein we have the following result. Let C^{⊗n} denote the n-th tensor power of the Carlitz module [2].

Theorem 1.2. If n > m, then Ext^{1}(C^{⊗m}, C^{⊗n}) has the structure of a t-module, and there is an exact sequence of t-modules

0 → C^{⊗(n−m)} → Ext^{1}(C^{⊗m}, C^{⊗n}) → L → 0,

where L is an m-dimensional iterated extension of 𝔾_{a}. Moreover, Ext_{0}^{1}(C^{⊗m}, C^{⊗n}) ≅ C^{⊗(n−m)}.

Since the (tractable) period π̃^{n−m} of C^{⊗(n−m)} is a power of the period π̃ of the Carlitz module (see D. Goss [12, Ch. 3]), one should compare Theorem 1.2 with the isomorphism

Ext_{MHS}^{1}(ℤ(m), ℤ(n)) ≅ ℂ/(2πi)^{n−m}ℤ, n > m,

from the theory of mixed Hodge structures [8].

The question of whether an analogue of the Weil-Barsotti formula holds for general t-modules is also interesting. Experimental evidence suggests that the general situation is subtle and that such formulas are not always valid for pure t-modules, e.g. in the form of Theorem 1.1, a Weil-Barsotti formula does not hold for C^{⊗n}.

The outline of this paper is as follows. In §2, we present definitions and fundamental results on extensions of t-modules. We prove Theorems 1.1 and 1.2 in §§3–4. In §5, we consider extensions of t-modules from an analytic viewpoint, so as to motivate the expectation that Ext^{1}(E, F) can be represented by a t-module for certain t-modules E and F. We consider the situation of elliptic curves in §6 and compare our results to a beautiful unpublished theorem of S. Lichtenbaum about extensions of elliptic curves over ℂ. We conclude in §7 with some remarks about extensions of t-motives.

2. Extensions of t-modules and biderivations

In this section we establish definitions and results about extensions of t-modules. For general definitions of t-modules we follow the terminology in Goss [12, Ch. 5]. We point out that the results in this section remain valid in the case that K is not perfect.

Let E and F be t-modules over K. An extension of E by F is a t-module X fitting into an exact sequence of t-modules

0 → F → X → E → 0. (1)

Then Ext^{1}(E, F) is defined to be the group (under Baer sum) of of t-module extensions of E by F up to Yoneda equivalence.

The main tool which enables us to compute this group is that of biderivations. The following definitions run parallel to those of W. D. Brownawell and the first author [6] and Gekeler [11], where extensions of t-modules by 𝔾_{a} were investigated.

Let Φ: 𝔽_{q}[t] → Mat_{d}(K){τ} and Ψ: 𝔽_{q}[t] → Mat_{e}(K){τ} be choices of coordinates for E and F respectively, where Mat_{d}(K){τ} is the ring of twisted polynomials with matrix coefficients. A (Φ, Ψ)-biderivation is an 𝔽_{q}-linear map

δ : 𝔽_{q}[t] → Mat_{e×d}(K){τ},

which satisfies

δ(ab) = Ψ(a)δ(b) + δ(a)Φ(b), ∀a, b ∈ 𝔽_{q}[t]. (2)

The 𝔽_{q}-vector space of all (Φ, Ψ)-biderivations is denoted Der(Φ, Ψ). It is straightforward to check that a biderivation δ is uniquely determined by the single value δ(t), and so if V ∈ Mat_{e×d}(K){τ}, we define δ_{V} ∈ Der(Φ, Ψ) to be that biderivation such that δ_{V} (t) = V. In this way, we have an isomorphism

V ↦ δ_{V} : Mat_{e×d}(K){τ} →^{∼} Der(Φ, Ψ),

of 𝔽_{q}-vector spaces. A biderivation δ is called inner if for some U ∈ Mat_{e×d}(K{τ}) we have

δ(a) = δ^{(U)}(a) := UΦ(a) − Ψ(a)U, ∀a ∈ 𝔽_{q}[t]. (3)

The subspace of Der(Φ, Ψ) of inner biderivations is denoted Der_{in}(Φ, Ψ).

Every (Φ, Ψ)-biderivation δ gives rise to an extension X = (𝔾_{a}^{d+e}, Υ) of E by F by defining

Υ(a) := (\begin{matrix}Φ(a) & 0 \\ δ(a) & Ψ(a) \end{matrix}) , ∀a ∈ 𝔽_{q}[t].

Again it is straightforward, using (2), to check that Υ is well-defined. Moreover, every extension of E by F defines a unique biderivation.

We note that if δ^{(U)} is an inner biderivation then in fact X is split. In this case the matrix Θ := ( \begin{matrix}I_{d} & 0 \\ U & I_{e} \end{matrix} ) provides the splitting, where I_{d}, I_{e} are identity matrices:

Θ^{−1}Υ(a)Θ = (\begin{matrix}Φ(a) & 0 \\ 0 & Ψ(a) \end{matrix}) , ∀a ∈ 𝔽_{q}[t].

Furthermore, it follows from the above discussion that every split extension arises in this way.

Suppose we are given two extensions of E by F which are Yoneda equivalent. It follows easily from the definition of Yoneda equivalence that the corresponding biderivations differ by an inner biderivation. It is straightforward to check that the (Baer) sum on Ext^{1}(E, F) corresponds to usual addition on the level of biderivations.

Now the endomorphisms of E and F induce (identical) 𝔽_{q}[t]-module structures on Ext^{1}(E, F). That is, if X represents a class in Ext^{1}(E, F) and b ∈ 𝔽_{q}[t], we can define two t-modules X · b and b · X, which ultimately represent the same class in Ext^{1}(E, F). Explicitly, suppose δ is the (Φ, Ψ)-biderivation corresponding to X and π: X → E is the natural map in (1). Let

X · b := ker ((e, x) ↦ Φ(b)(e) − π(x) : E ⊕ X → E) .

Then X · b is itself a t-module extension of E by F, and the operation of 𝔽_{q}[t] on it is given by

(Υ · b)(a) = (\begin{matrix}Φ(a) & 0 \\ δ(a)Φ(b) & Ψ(a) \end{matrix}) , ∀a ∈ 𝔽_{q}[t].

On the other hand, we can similarly use endomorphisms of F to define an extension b · X whose t-module structure is given by

(b · Υ)(a) = (\begin{matrix}Φ(a) & 0 \\ Ψ(b)δ(a) & Ψ(a) \end{matrix}) , ∀a ∈ 𝔽_{q}[t].

To see that X · b and b · X are equivalent extensions, we note that ε: 𝔽_{q}[t] → Mat_{e×d}(K){τ} defined by

ε(a) := δ(a)Φ(b) − Ψ(b)δ(a), ∀a ∈ 𝔽_{q}[t],

is in fact the inner biderivation δ^{(U)}, with U = δ(t) in (3). That is,

b: δ(·) ↦ δ(·)Φ(b) and b: δ(·) ↦ Ψ(b)δ(·) (4)

define 𝔽_{q}[t]-module structures on Der(Φ, Ψ) which are the same modulo Der_{in}(Φ, Ψ). We record the results from the preceding paragraphs in the following lemma.

Lemma 2.1. Let E = (𝔾_{a}^{d}, Φ) and F = (𝔾_{a}^{e}, Ψ) be t-modules. Then

Ext^{1}(E, F) ≅ Der(Φ, Ψ)/ Der_{in}(Φ, Ψ)

as 𝔽_{q}[t]-modules.

For U ∈ Mat_{d_{1}×d_{2}}(K){τ}, we let dU ∈ Mat_{d_{1}×d_{2}}(K) be the constant term of U as a polynomial in τ, and we define the following subspaces of Der(Φ, Ψ):

Der_{0}(Φ, Ψ) := {δ ∈ Der(Φ, Ψ) : dδ(t) = 0},
Der_{si}(Φ, Ψ) := {δ^{(U)} ∈ Der_{in}(Φ, Ψ) : dU = 0}.

Note that Der_{0} differs here from definitions in [6] and [11]. The utility of Der_{0}(Φ, Ψ) is derived from the following lemma, whose immediate corollary follows from Lemma 2.1. Biderivations in Der_{si}(Φ, Ψ) are called strictly inner, and clearly Der_{si}(Φ, Ψ) ⊂ Der_{0}(Φ, Ψ). We will study Der_{si}(Φ, Ψ) in more detail in §5.

The map dΦ: 𝔽_{q}[t] → Mat_{d}(K) defines a t-module whose underlying space is the tangent space Lie(E) ≅ K^{d}. Furthermore, the map δ ↦ dδ: Der(Φ, Ψ) → Der(dΦ, dΨ) is 𝔽_{q}-linear, and it is 𝔽_{q}[t]-linear modulo inner biderivations.

Lemma 2.2. Let E = (𝔾_{a}^{d}, Φ) and F = (𝔾_{a}^{e}, Ψ) be t-modules. The following is an exact sequence of 𝔽_{q}[t]-modules:

(5) 0 → \frac{Der_{0}(Φ, Ψ)}{Der_{0}(Φ, Ψ) ∩ Der_{in}(Φ, Ψ)} → \frac{Der(Φ, Ψ)}{Der_{in}(Φ, Ψ)} → \frac{Der(dΦ, dΨ)}{Der_{in}(dΦ, dΨ)}.

If θ ∈ K is transcendental over 𝔽_{q}, then the final map in this sequence is surjective.

We define Ext_{0}^{1}(E, F) to be the 𝔽_{q}[t]-submodule of Ext^{1}(E, F) corresponding to Der_{0}(Φ, Ψ)/(Der_{0}(Φ, Ψ) ∩ Der_{in}(Φ, Ψ)).

Corollary 2.3. The sequence in Lemma 2.2 corresponds to an exact sequence of 𝔽_{q}[t]-modules,

0 → Ext_{0}^{1}(E, F) → Ext^{1}(E, F) → Ext^{1}(Lie(E), Lie(F)),

where the final map is surjective if θ ∈ K is transcendental over 𝔽_{q}.

Proof of Lemma 2.2. Injectivity on the left of (5) is clear. To show exactness in the center, first any δ ∈ Der_{0}(Φ, Ψ) maps to 0 in Der(dΦ, dΨ). On the other hand, suppose δ ∈ Der(Φ, Ψ) and dδ ∈ Der(dΦ, dΨ) is inner, say dδ(t) = UdΦ(t) − dΨ(t)U with U ∈ Mat_{e×d}(K){τ}. Then δ − δ_{Φ,Ψ}^{(U)} ∈ Der_{0}(Φ, Ψ) represents the same class as δ in Der(Φ, Ψ)/ Der_{in}(Φ, Ψ).

In the case that θ ∈ K is transcendental over 𝔽_{q}, we show surjectivity on the right. We suppose dΦ(t) = I_{d}θ + M and dΨ(t) = I_{e}θ + N, where M and N are nilpotent. Without loss of generality we can assume M and N are both upper triangular. The spaces of biderivations Der(dΦ, dΨ) and Der_{in}(dΦ, dΨ) are both naturally K-linear, and the map defining elements of Der_{in}(dΦ, dΨ),

(6) U ↦ δ_{dΦ,dΨ}^{(U)}(t) : Mat_{e×d}(K){τ} → Mat_{e×d}(K){τ},

is K-linear and respects the grading by degrees in τ. Now

δ_{dΦ,dΨ}^{(U)}(t) = Uθ − θU + UM − NU. (7)

If U_{ij} is the matrix in Mat_{e×d}(K) with a 1 in the ij-th entry and zeros elsewhere, then {U_{m1}τ^{r}, U_{m−1,1}τ^{r}, … } = {U_{ij}τ^{r}}_{i=m,...,e}^{j=1,...,d} is an ordered K-basis for Mat_{e×d}(K)τ^{r}. Using this basis, it follows from (7) and the fact that M and N are both upper triangular that the map in (6), restricted to Mat_{e×d}(K)τ^{r}, is lower triangular with θ^{q^{r}} − θ along the diagonal. Since θ^{q^{r}} − θ is non-zero by our assumption on θ,

U ↦ δ_{dΦ,dΨ}^{(U)}(t) : Mat_{e×d}(K){τ}τ →^{∼} Mat_{e×d}(K){τ}τ,

is an isomorphism of K-vector spaces. Therefore, for every δ ∈ Der(dΦ, dΨ), if dδ(t) = 0, then δ is inner, and so the right-hand side of (5) is surjective.

3. A Weil-Barsotti formula

In this section, we prove Theorem 1.1. Let E = (𝔾_{a}, Φ) be a Drinfeld module of rank r ≥ 2, where Φ(t) = θ + a_{1}τ + ⋯ + a_{r}τ^{r}. For an extension X of E by C, we let δ_{X} ∈ Der(Φ, C) be its associated biderivation.

Proof of Theorem 1.1(a). An extension X of E by C splits if and only if δ_{X} ∈ Der_{in}(Φ, C), i.e. if there exists u ∈ K{τ} such that δ_{X}(t) = uΦ(t) − (θ + τ)u. If u = cτ^{m}, then we obtain an inner biderivation δ^{(u)} such that

\begin{aligned}δ^{(u)}(t) & = uΦ(t) − (θ + τ)u \\ & = cθ^{q^{m}}τ^{m} + ca_{1}^{q^{m}}τ^{m+1} + ⋯ + ca_{r}^{q^{m}}τ^{m+r} − θcτ^{m} − c^{q}τ^{m+1}. \end{aligned} (8)

If n := deg_{τ}(δ_{X}(t)) is greater than r − 1, we can repeatedly subtract (8) from δ_{X}(t), with m = n − r, n − r − 1, …, 0, to reduce the τ-degree of δ_{X}(t); eventually this degree will be < r. Namely, any extension X is equivalent to an extension X′ with deg_{τ}(δ_{X′}(t)) ≤ r − 1, which we call the reduced representative of X.

According to (8), the non-zero inner biderivations of least degree have degree r, so two extensions Z and Y satisfying deg_{τ}(δ_{Z}(t)) ≤ r − 1, deg_{τ}(δ_{Y} (t)) ≤ r − 1, and δ_{Z}(t) ≠ δ_{Y} (t), are inequivalent. Therefore the map

X ↦ δ_{X′}(t) : Ext^{1}(E, C) →^{∼} V := {b_{0} + b_{1}τ + ⋯ + b_{r−1}τ^{r−1} : b_{i} ∈ K}, (9)

where X′ is the reduced representative of X, induces an isomorphism of 𝔽_{q}[t]-modules.

We now turn to the t-module structure on Ext^{1}(E, C). Recall from (4) that multiplication-by-t on Ext^{1}(E, C) is defined by tα = (θ + τ)α = αΦ(t). Here we think of α as being an element of V. In order to see the action explicitly, it is enough to consider α = b_{i}τ^{i}:

\begin{aligned}t(b_{0}) & = (θ + τ)b_{0} = θb_{0} + b_{0}^{q}τ, \\ t(b_{1}τ) & = (θ + τ)b_{1}τ = θb_{1}τ + b_{1}^{q}τ^{2}, \\ ⋮ & ⋮ \\ t(b_{r−1}τ^{r−1}) & = (θ + τ)b_{r−1}τ^{r−1} = θb_{r−1}τ^{r−1} + b_{r−1}^{q}τ^{r}. \end{aligned} (10)

Using (8) with m = 0, c = b_{r−1}^{q}/a_{r}, we can rewrite the last identity of (10) as

t(b_{r−1}τ^{r−1}) = (^{\frac{b_{r−1}^{q^{2}}}{a_{r}^{q}}} − \frac{b_{r−1}^{q}a_{1}}{a_{r}})τ − \frac{b_{r−1}^{q}}{a_{r}} (a_{2}τ^{2} + ⋯ + a_{r−2}τ^{r−2}) + (θb_{r−1} − \frac{b_{r−1}^{q}a_{r−1}}{a_{r}})τ^{r−1}.

Thus in terms of the elements e_{i} = τ^{i} (i = 0, …, r − 1) of V, the t-module structure on Ext^{1}(E, C) can be expressed by the map Π: 𝔽_{q}[t] → Mat_{r}(K){τ}, defined by

Π(t) := (\begin{matrix}θ & 0 & ⋯ & ⋯ & 0 & 0 \\ τ & θ & 0 & & ⋮ & −\frac{a_{1}}{a_{r}} τ + \frac{1}{a_{r}^{q}} τ^{2} \\ 0 & τ & θ & 0 & ⋮ & −\frac{a_{2}}{a_{r}} τ \\ ⋮ & 0 & τ & θ & 0 & ⋮ \\ ⋮ & & 0 & τ & θ & −\frac{a_{r−2}}{a_{r}} τ \\ 0 & ⋯ & ⋯ & 0 & τ & θ − \frac{a_{r−1}}{a_{r}} τ. \end{matrix}) .

Comparing with Taguchi in [19, §5], it is clear that the t-module Ext^{1}(E, C) is an extension of 𝔾_{a} by the t-module denoted there Ě. By Corollary 2.3 and the characterization of V in (9), it is clear Ext^{1}(E, C) is an extension of 𝔾_{a} by E^{∨}. Thus E^{∨} is the same as Taguchi’s t-module. Moreover, Taguchi shows [19, Thm. 5.1] that E^{∨} is a pure t-module isomorphic to E^{∧(r−1)}.

We turn to part (b), and for simplicity we assume that a_{r} = 1; the general case follows similarly. The t-module structure on E^{∨} is then defined by

Ψ(t) := (\begin{matrix}θ & 0 & ⋯ & 0 & −a_{1}τ + τ^{2} \\ τ & θ & 0 & ⋮ & −a_{2}τ \\ 0 & τ & θ & 0 & ⋮ \\ ⋮ & 0 & τ & θ & −a_{r−2}τ \\ 0 & ⋯ & 0 & τ & θ − a_{r−1}τ. \end{matrix}) .

Proof of Theorem 1.1(b). Consider the biderivation δ_{X}: 𝔽_{q}[t] → Mat_{1×(r−1)}(K){τ} of any extension X of E^{∨} by C. The inner biderivations are of the form δ^{(U)} = UΨ − (θ + τ)U for U = (u_{i}) ∈ Mat_{1×(r−1)}(K){τ}. Explicitly, v = (v_{i}) ∈ Mat_{1×(r−1)}(K){τ} defines an inner biderivation, δ^{(U)} = δ_{v}, if it has the form

\begin{aligned}v_{i} & = u_{i}θ − θu_{i} + u_{i+1}τ − τu_{i} (1 ≤ i ≤ r − 2), \\ v_{r−1} & = u_{r−1}θ − θu_{r−1} + u_{1}τ^{2} − τu_{r−1} − ∑_{j=1}^{j=r−1} u_{j}a_{j}τ. \end{aligned}

An inner biderivation δ_{v} is said to be basic if u_{i} = 0 for all i ≠ s and u_{s} = cτ^{m} with c ∈ K; we write v = v(s, c, m) = (v_{1}, …, v_{r−1}). Explicitly written, these are

\begin{aligned}v(1, c, m) & = (cθ^{q^{m}}τ^{m} − cθτ^{m} − c^{q}τ^{m+1}, 0, …, 0, cτ^{m+2} − ca_{1}^{q^{m}}τ^{m+1}), \\ v(2, c, m) & = (cτ^{m+1}, cθ^{q^{m}}τ^{m} − cθτ^{m} − c^{q}τ^{m+1}, 0, …, 0, −ca_{2}^{q^{m}}τ^{m+1}), \\ v(z, c, m) & = (0, …, cτ^{m+1}, cθ^{q^{m}}τ^{m} − cθτ^{m} − c^{q}τ^{m+1}, 0, …, 0, −ca_{z}^{q^{m}}τ^{m+1}), \\ v(r − 1, c, m) & = (0, …, 0, cτ^{m+1}, cθ^{q^{m}}τ^{m} − cθτ^{m} − c^{q}τ^{m+1} − ca_{r−1}^{q^{m}}τ^{m+1}), \end{aligned} (11)

where for 2 ≤ z ≤ r − 2 the possible non-zero coordinates of v(z, c, m) are v_{z−1}, v_{z}, and v_{r−1}. Every inner biderivation arises from an additive combination of basic δ_{v}.

Consider G ⊂ Mat_{1×(r−1)}(K){τ} consisting of elements u := (u_{i}) with τ-degrees of u_{1}, …, u_{r−2} zero and the τ-degree of u_{r−1} less than one. In other words, u_{1}, …, u_{r−2} ∈ K and u_{r−1} = c + dτ with c, d ∈ K. Elements of G give rise to biderivations which we will call reduced.

Lemma 3.1. The map u ↦ δ_{u}: G → Der(Ψ, C) induces an isomorphism

G ≅ \frac{Der(Ψ, C)}{Der_{in}(Ψ, C)} ≅ Ext^{1}(E^{∨}, C)

of 𝔽_{q}-vector spaces.

Proof of Lemma. We need only to prove the first isomorphism by Lemma 2.1. Let X be an extension. We want to subtract appropriate v’s in (11) from δ_{X}(t) = u = (u_{1}, u_{2}, …, u_{r−1}) so that the resulting biderivation is reduced. In this process, we need to keep track of the τ-degrees of the u_{j}’s. We define the τ-degree of u to be the vector d_{u} := (d_{1}(u), …, d_{r−1}(u)) with d_{j}(u) = τ-degree of u_{j}. Given two vectors d and d′ with integer coefficients, we shall say d ≤ d′ if d_{i} ≤ d′_{i} for all i. So our claim is that u can be reduced to a biderivation ũ such that dũ ≤ (0, …, 0, 1).

Let n be the maximum of the integers d_{j}(u); one has d_{j}(u) ≤ n for all j. We can modify u by v(2, c, d_{1} − 1) for an appropriate c ∈ K to obtain u′ such that d_{1}(u′) < d_{1}(u) and d_{g}(u′) ≤ n for g ≥ 2. Subtracting an appropriate v(3, c, d_{2}(u′)), we obtain u′′ such that d_{2}(u′′) < d_{2}(u′) and d_{g}(u′′) ≤ n for g ≥ 3. Repeating this for z = 4, …, r − 2 using appropriate v(z, c, m), we obtain a w ∈ Mat_{1×(r−1)}(K){τ} whose degree vector is less than or equal to (n − 1, n − 1, …, n − 1, n).

If n ≤ 1, we are done. If not (n ≥ 2), we can subtract an appropriate v(1, c, n − 2) to obtain a vector w whose degree vector is less than or equal to (n − 1, n − 1, …, n − 1, n − 1). We repeat the procedure in the two paragraphs above until we arrive at a vector w whose degree vector is less than or equal to (0, …, 0, 1).

We can now determine the t-module structure on Ext^{1}(E^{∨}, C). By the previous lemma, it suffices to see this structure on G. Consider the elements e_{i} ∈ G ⊂ Mat_{1×(r−1)}(K){τ}, 1 ≤ i ≤ r, defined as follows. For 1 ≤ i ≤ r − 1, we take e_{i} to be the vector with 1 in the i-th coordinate and zeros elsewhere. We take e_{r} to be the vector with τ in the last coordinate and zeros elsewhere. The structure of a t-module on G is completely described by the action of t on elements of the form be_{i}, b ∈ K, since additive combinations of such elements give all of G.

Consider t(be_{i}) = (θ + τ)be_{i} = θe_{i} + b^{q}τe_{i}. The last is no longer an element of G, and we need its equivalent vector in G. Using the reduction procedure of Lemma 3.1, it is easily computed that

\begin{aligned}(θ + τ)(be_{i}) & = θbe_{n} + (∑_{f=0}^{r−i−1}b^{q^{r−i−f}} a_{r−f})e_{r}, 1 ≤ i ≤ r − 1, \\ (θ + τ)(be_{r−1}) & = θbe_{r−1} + b^{q}e_{r} \\ (θ + τ)(be_{r−2}) & = θbe_{r−2} + (b^{q^{2}} + b^{q}a_{r−1})e_{r} \\ (θ + τ)(be_{r}) & = (θ + ∑_{f=1}^{r}b^{q^{f}} a_{f})e_{r}. \end{aligned}

Thus the t-module structure Ξ: 𝔽_{q}[t] → Mat_{r}(K){τ} on Ext^{1}(E^{∨}, C) is completely described by

Ξ(t) := (\begin{matrix}θ & 0 & ⋯ & ⋯ & 0 \\ 0 & θ & ... & & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & & ... & θ & 0 \\ α_{1} & α_{2} & ⋯ & α_{r−1} & α_{r} \end{matrix}) .

Here α_{r} = θ + a_{1}τ + a_{2}τ^{2} + ⋯ + a_{r−1}τ^{r−1} + τ^{r}, and the others are given by α_{n} = ∑_{f=0}^{r−n−1} τ^{r−n−f}a_{r−f}, for 1 ≤ n ≤ r − 1. Moreover, Ext^{1}(E^{∨}, C) is an extension of 𝔾_{a}^{r−1} by E. By Lemma 3.1, (E^{∨})^{∨} is 1-dimensional, which completes the proof.

4. Extensions of tensor powers of the Carlitz module

In this section we prove Theorem 1.2. Recall that the n-th tensor power of the Carlitz module is the n-dimensional pure t-module C^{⊗n}: 𝔽_{q}[t] → Mat_{n}(K){τ} defined by

C^{⊗n}(t) := (\begin{matrix}θ & 1 & 0 \\ ⋮ & ... & 1 \\ τ & ⋯ & θ \end{matrix}) .

That is, C^{⊗n}(t) = θI_{n} + N_{n} + E_{n}τ, where

N_{n} := (\begin{matrix}0 & 1 & 0 \\ ⋮ & ... & 1 \\ 0 & ⋯ & 0 \end{matrix}) and E_{n} := (\begin{matrix}0 & ⋯ & 0 \\ ⋮ & & ⋮ \\ 1 & ⋯ & 0 \end{matrix}) .

See Anderson and D. S. Thakur [2] for more details.

Fix m < n. The following lemma determines representatives for elements of Der(C^{⊗m}, C^{⊗n})/ Der_{in}(C^{⊗m}, C^{⊗n}).

Lemma 4.1. Let

G := \{(\begin{matrix}∗ & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ ∗ & 0 & ⋯ & 0 \end{matrix}) ∈ Mat_{n×m}(K)\} .

Then V ↦ δ_{V} : G →^{∼} Der(C^{⊗m}, C^{⊗n})/ Der_{in}(C^{⊗m}, C^{⊗n}) is an isomorphism of 𝔽_{q}- vector spaces.

Proof. Let Q_{ij} be the n × m matrix with a 1 in the ij-th entry and zeros elsewhere. For c ∈ K and k ≥ 0, by taking cQ_{ij}τ^{k} for U in δ^{(U)}(t), we define

\begin{aligned}δ_{ijk}^{[c]} & := (cQ_{ij}τ^{k})C^{⊗m}(t) − C^{⊗n}(t)(cQ_{ij}τ^{k}) \\ & = c(θ^{q^{k}} − θ)Q_{ij}τ^{k} + c(Q_{ij}N_{m} − N_{n}Q_{ij})τ^{k} + (cQ_{ij}E_{m} − c^{q}E_{n}Q_{ij})τ^{k+1}. \end{aligned} (12)

Biderivations arising from (12) generate Der_{in}(C^{⊗m}, C^{⊗n}) as an 𝔽_{q}-vector space.

Suppose that V = (v_{ij}) ∈ Mat_{n×m}(K{τ}) is arbitrary and that deg_{τ}(v_{ij}) ≤ r for some r ≥ 1. We will show that by subtracting matrices in (12) from V in various ways we can replace V by a matrix V′ which has each entry of τ-degree ≤ r − 1 and also δ_{V′} equivalent to δ_{V} modulo Der_{in}(C^{⊗m}, C^{⊗n}).

We bootstrap our way through the entries of V in the following way. Let

D_{ij} := {δ_{ijk}^{[c]} : k ≥ 0, c ∈ K}.

Define a function F from the set of subsets of I := {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ m} to itself. We set F (S) to be those entries of V whose degrees in τ can be decreased by subtracting an element of ∑_{(i,j)∈S} D_{ij}, without increasing the degrees of the other entries. Our claim then is that F (I) = I. The following containments can be easily checked:

\begin{aligned}F (\{2 ≤ i ≤ n; j = m\}) & ⊃ & \{2 ≤ i ≤ n; j = 1\}, \\ F (\{2 ≤ i ≤ n; j = m\} ∪ \{i − j ≥ 2\}) & ⊃ & \{i − j ≥ 1\}, \\ F (\{j = m\} ∪ \{i − j ≥ 2\}) & ⊃ & \{i − j ≥ 1\} ∪ \{(1, 1), (1, m)\}, \\ F (\{j = m\} ∪ \{i − j ≥ 1\}) & ⊃ & \{i − j ≥ 0\} ∪ \{(1, m)\}, \\ F (\{j = m\} ∪ \{i − j ≥ −ℓ\}) & ⊃ & \{i − j ≥ −ℓ − 1\} ∪ \{(1, m)\}, \end{aligned} (13)

where the last containment holds for all ℓ = 0, …, m − 1.

Therefore, we can assume that every entry of V is a constant from K. Now that the τ-degree of each entry of V is 0, one checks the containments in (13) still hold with the exception that all sets on the right-hand side must have {j = 1} removed. That is, we can adjust V so that it can be replaced by a matrix in G, but elements of G can be reduced no further.

Proof of Theorem 1.2. Let G be given as in Lemma 4.1 so that by Lemma 2.1

G ≅ Ext^{1}(C^{⊗m}, C^{⊗n}).

Recalling the definition of Q_{ij} from the proof above, let

e_{1} := Q_{11}, …, e_{n} := Q_{n1},

be basis vectors for G over K. Combining Lemmas 2.1 and 4.1, G has a natural 𝔽_{q}[t]-module structure which we now make explicit. For c ∈ K,

t · (ce_{1}) = C^{⊗n}(t)(ce_{1}) = θce_{1} + c^{q}τe_{n}, (14)

and

t · (ce_{i}) = C^{⊗n}(t)(ce_{i}) = θce_{i} + ce_{i−1}, 2 ≤ i ≤ n. (15)

We note that as defined in (12),

∑_{i=0}^{m−1} δ_{n−i,m−i,0}^{[c^{q}]} = c^{q}Q_{n1}τ − c^{q}Q_{n−m,1} = c^{q}τe_{n} − c^{q}e_{n−m},

and since this sum defines an inner (C^{⊗m}, C^{⊗n})-biderivation, we subtract it from (14) and find that

t · (ce_{1}) = θce_{1} + c^{q}e_{n−m}.

Therefore, combining this with (15), we see that the 𝔽_{q}[t]-module structure on G can be expressed as a t-module by the map Π: 𝔽_{q}[t] → Mat_{n}(K){τ} defined by

Π(t) := ( \begin{matrix}C^{⊗(n−m)}(t) & \begin{matrix}0 & ⋯ & 0 \\ ⋮ & & ⋮ \\ 1 & ⋯ & 0 \end{matrix} \\ 0 & I_{m}θ + N_{m} \end{matrix} ) .

Thus there is an exact sequence of t-modules,

0 → C^{⊗(n−m)} → G → L → 0,

where L is an m-dimensional iterated extension of 𝔾_{a}, and one checks that moreover C^{⊗(n−m)} = Ext_{0}^{1}(C^{⊗m}, C^{⊗n}).

5. Periods of t-modules and extensions

In this section we would like to motivate the expectation that Ext^{1}(E, F), for tmodules E and F, can be given the structure of a t-module. Our insights here come from the analytic theory of t-modules, and our main tool will be generalizations of quasi-periodic functions defined in [6] and [11] for extensions by 𝔾_{a}. Moreover we study an analogue of the de Rham map of Gekeler [11]. For more details on the general analytic theory of t-modules see Goss [12, Ch. 5].

Let 𝕂 be the completion of the algebraic closure of the Laurent series field 𝔽_{q}((1/θ)), where θ is an independent variable. Let ι: 𝔽_{q}[t] → 𝕂 be defined by t ↦ θ.

Let E = (𝔾_{a}^{d}, Φ) and F = (𝔾_{a}^{e}, Ψ) be uniformizable t-modules over 𝕂. We take Exp_{E}: Lie(E)(𝕂) → E(𝕂) to be the exponential map of E. Because E is uniformizable, there is an exact sequence of 𝔽_{q}[t]-modules

0 → Λ_{Φ} → 𝕂^{d} →^{Exp_{Φ}} 𝕂^{d} → 0,

where 𝔽_{q}[t] operates by dΦ on the central 𝕂^{d} and by Φ on the right one. Also, Λ_{Φ} is the period lattice of Φ and is a discrete 𝔽_{q}[t]-submodule of 𝕂^{d}. Similarly, we define Exp_{Ψ} and Λ_{Ψ} for F.

Lemma 5.1. For each δ ∈ Der_{0}(Φ, Ψ) there is a unique 𝔽_{q}-linear entire function F_{δ}: 𝕂^{d} → 𝕂^{e} such that for z ∈ 𝕂^{d},

F_{δ}(z) ≡ 0 (mod deg q),
F_{δ}(dΦ(a)z) = Ψ(a)F_{δ}(z) + δ(a) Exp_{Φ}(z), ∀a ∈ 𝔽_{q}[t].

Proof. The proof here is essentially the same as the that of the existence of the exponential function, and in particular we can easily adapt the proof of Proposition 2.1.4 in Anderson [1] to this situation.

Lemma 5.2. Suppose δ^{(U)} ∈ Der_{0}(Φ, Ψ) ∩ Der_{in}(Φ, Ψ). Then

F_{δ}(z) = U Exp_{Φ}(z) − Exp_{Ψ}(dU · z).

Proof. This follows directly from Lemma 5.1, using the fact that δ^{(U)} ∈ Der_{0}(Φ, Ψ) if and only if dU dΦ(a) = dΨ(a) dU for all a ∈ 𝔽_{q}[t].

If X = (𝔾_{a}^{d+e}, Υ) is an extension of E by F defined by a biderivation δ ∈ Der_{0}(Φ, Ψ), then Lemma 5.1 and the uniqueness of the exponential function imply that

Exp_{Υ} (\begin{matrix}z \\ u \end{matrix}) = (\begin{matrix}Exp_{Φ}(z) \\ Exp_{Ψ}(u) + F_{δ}(z) \end{matrix})

is the exponential function for X. Also, since E and F are both uniformizable, so is X. Moreover, the period lattice of X is

Λ_{Υ} = \{(\begin{matrix}λ \\ η \end{matrix}) : λ ∈ Λ_{Φ}, Exp_{Ψ}(η) + F_{δ}(λ) = 0\} .

For each (λ, η) ∈ Λ_{Υ}, it follows that (λ, η+dΨ(a)μ) ∈ Λ_{Υ} for all a ∈ 𝔽_{q}[t] and μ ∈ Λ_{Ψ}.

Proposition 5.3. The following map is a homomorphism of 𝔽_{q}[t]-modules:

DR := DR_{(Φ,Ψ)} : Der_{0}(Φ, Ψ)/ Der_{si}(Φ, Ψ) → Hom_{𝔽_{q}[t]}(Λ_{Φ}, Ψ(𝕂)),
δ  ↦  (λ ↦ F_{δ}(λ)).

Proof. The map DR_{0} : δ ↦ (λ ↦ F_{δ}(λ)) on Der_{0}(Φ, Ψ) → Hom_{𝔽_{q}[t]}(Λ_{Φ}, Ψ(𝕂)) is clearly well-defined and 𝔽_{q}-linear by Lemma 5.1. If δ^{(U)} ∈ Der_{si}(Φ, Ψ), then by definition dU = 0, and so by Lemma 5.2, DR_{0}(δ^{(U)}) = 0. Therefore DR is welldefined. Furthermore, from (4) and Lemma 5.1, it follows that

F_{a·δ}(z) = F_{Ψ(a)δ(·)}(z) = Ψ(a)F_{δ}(z), ∀a ∈ 𝔽_{q}[t],

and so DR(a · δ)(λ) = Ψ(a)(DR(δ)(λ)), for all a ∈ 𝔽_{q}[t].

Remark 5.4. In the case of extensions of Drinfeld modules by 𝔾_{a}, the map DR specializes to the de Rham homomorphism of Gekeler [11]. Gekeler shows in this case that the de Rham map is in fact an isomorphism. In general, determining the kernel and cokernel of DR is a delicate matter and will require further study. However, Theorems 1.1 and 1.2 can be used to imply that the de Rham maps in their respective situations are indeed isomorphisms. A straightforward modification of the proof of [11, Thm. 3.1] yields the following partial result.

Proposition 5.5. Suppose E = (𝔾_{a}, Φ) and F = (𝔾_{a}, Ψ) are Drinfeld modules of ranks r and s respectively. If r > s, then DR_{(Φ,Ψ)} is injective.

For the remainder of this section we will consider the implications of the assumption that DR is an isomorphism. In this way we attempt to motivate the idea that Ext_{0}^{1}(E, F) can be given the structure of a t-module.

Suppose 𝕂^{d} is an 𝔽_{q}[t]-module defined by an 𝔽_{q}-linear representation ϕ: 𝔽_{q}[t] → Mat_{d}(𝕂) such that ϕ(t) = θI_{d} + N with N nilpotent. If Λ is any finitely generated discrete 𝔽_{q}[t]-submodule of 𝕂^{d} of rank r, then following the language of Anderson [1, §4.4], we call 𝕂^{d}/Λ a t-torus of dimension d and rank r. Through its exponential function, every uniformizable t-module is isomorphic to a t-torus.

Let λ_{1}, …, λ_{r} be an 𝔽_{q}[t]-basis for Λ_{Φ}. Choosing this basis amounts to fixing isomorphisms

(16) Hom_{𝔽_{q}[t]}(Λ_{Φ}, Ψ(𝕂)) ≅ Ψ(𝕂)^{r} ≅ (𝕂^{e})^{r}/(Λ_{Ψ})^{r}.

We let

(17) W := W_{(Φ,Ψ)} := \{(Uλ_{1}, …, Uλ_{r}) ∈ Mat_{e×r}(𝕂) : \begin{matrix}U ∈ Mat_{e×d}(𝕂), \\ UdΦ(t) = dΨ(t)U \end{matrix} \} .

Suppose δ^{(U)} ∈ Der_{0}(Φ, Ψ) ∩ Der_{in}(Φ, Ψ) for U ∈ Mat_{e×d}(𝕂). By Lemma 5.2 and (16),

DR(δ^{(U)}) = (Exp_{Ψ}(Uλ_{1}), …, Exp_{Ψ}(Uλ_{r})) ∈ Mat_{e×r}(𝕂),

and so DR(Der_{0}(Φ, Ψ) ∩ Der_{in}(Φ, Ψ)) = Exp_{Ψ}^{⊕r}(W). Thus by Corollary 2.3, DR_{(Φ,Ψ)} induces a homomorphism of 𝔽_{q}[t]-modules,

(18) DR_{(Φ,Ψ)} : Ext_{0}^{1}(E, F) → Mat_{e×r}(𝕂)/(W + Λ_{Ψ}^{r}),

which is an isomorphism if the original DR_{(Φ,Ψ)} is one. Depending on E and F, the right-hand side may or may not be isomorphic (rigid analytically) to a t-torus. We consider the following situations.

Example 5.6. Suppose E = (𝔾_{a}, Φ) and F = (𝔾_{a}, Ψ) are Drinfeld modules of rank 2 and s respectively. Let λ_{1}, λ_{2} and μ_{1}, …, μ_{s} be generators for their period lattices over 𝔽_{q}[t]. The exact sequence

0 → W → 𝕂^{2} →^{f} 𝕂 → 0,

where f(x, y) = λ_{1}y − λ_{2}x, provides a choice of coordinates on the right-hand side of (18), and f(Λ_{Ψ} ⊕ Λ_{Ψ}) is the 𝔽_{q}[t]-submodule generated by {λ_{i}μ_{j}}_{i=1,2}^{j=1,...,s}.

If s = 1, then Proposition 5.5 implies that Ext_{0}^{1}(E, F) ↪ 𝕂/(𝔽_{q}[t]λ_{1}μ_{1}+𝔽_{q}[t]λ_{2}μ_{1}), and Theorem 1.1 (and also Woo [20, Prop. 7]) confirms that this is an isomorphism of t-tori. If s ≥ 2, then it is easy to construct examples where f(Λ_{Ψ} ⊕ Λ_{Ψ}) is not discrete in 𝕂, and so in such cases the right-hand side of (18) is not a t-torus (cf. Theorem 6.1).

Proposition 5.7. Suppose E = (Φ, 𝔾_{a}) and F = (Ψ, 𝔾_{a}) are Drinfeld modules of rank r and s respectively, with r > s. If DR_{(Φ,Ψ)} is an isomorphism, then Ext_{0}^{1}(E, F) is isomorphic as an 𝔽_{q}[t]-module to a t-torus of dimension r − 1 and rank rs.

Proof. Let λ_{1}, …, λ_{r} and μ_{1}, …, μ_{s} be 𝔽_{q}[t]-bases for Λ_{Φ} and Λ_{Ψ} respectively. Because E and F are both 1-dimensional, Der_{in}(Φ, Ψ) ⊂ Der_{0}(Φ, Ψ). As in (17),

W = W_{(Φ,Ψ)} = {(Uλ_{1}, …, Uλ_{r}) ∈ Mat_{1×r}(𝕂) : U ∈ 𝕂}.

We claim that W ∩ (Λ_{Ψ})^{r} = {0}. Suppose w ∈ W ∩ (Λ_{Ψ})^{r}. Then

w = (Uλ_{1}, …, Uλ_{r}) = (ν_{1}, …, ν_{r}), ν_{i} ∈ Λ_{Ψ}.

Because r > s, there is a non-trivial 𝔽_{q}[t]-linear dependency ∑ dΨ(a_{i})ν_{i} = 0, and thus U · ∑ dΦ(a_{i})λ_{i} = 0. Since λ_{1}, …, λ_{r} are linearly independent over 𝔽_{q}[t], it follows that U = 0.

By the following argument, the image of Λ_{Ψ}^{r} is discrete in Mat_{1×r}(𝕂)/W. We proceed by induction on r, for which the base case (r = 2) is trivial. Furthermore, it suffices to continue with r = s + 1, which we will now assume; the cases where r > s + 1 follow as straightforward consequences. The 𝕂-linear map

\begin{aligned}f : & Mat_{1×r}(𝕂) & → & Mat_{1×(r−1)}(𝕂), \\ & (x_{1}, …, x_{r}) & ↦ & (λ_{2}x_{1} − λ_{1}x_{2}, …, λ_{r}x_{r−1} − λ_{r−1}x_{r}), \end{aligned}

has kernel W. We need to show that V := 𝔽_{q}((1/θ)) · f(Λ_{Ψ}) ⊂ Mat_{1×(r−1)}(𝕂) is an rs-dimesional vector space over 𝔽_{q}((1/θ)). The typical element of V has the form

ν = (μ_{1}, …, μ_{s}) A (\begin{matrix}λ_{2} & ⋯ & 0 \\ −λ_{1} & ... & ⋮ \\ ⋮ & ... & λ_{r} \\ 0 & ⋯ & −λ_{r−1} \end{matrix}) , (19)

where A = (α_{ij}) ∈ Mat_{s×r}(𝔽_{q}((1/θ))). Let B denote the r × (r − 1) matrix in the above formula. By our assumption that s = r − 1, the matrix AB is square, and its determinant is

det(AB) = λ_{2} ⋯ λ_{r−1} ∑_{j=1}^{r}(−1)^{r−j}λ_{j} det(A_{j}),

where A_{j} is the s × s minor of A with the j-th column removed. If ν = 0, then this implies that det(AB) = 0, and since λ_{1}, …, λ_{r} are 𝔽_{q}((1/θ))-linearly independent, det(A_{j}) = 0 for each j. Thus the rank of the matrix A is less than s, and we can rewrite the last row of A as an 𝔽_{q}((1/θ))-linear combination of the other rows, say α_{sj} = ∑_{i=1}^{s−1} β_{i}α_{ij} with β_{i} ∈ 𝔽_{q}((1/θ)). The formulation in (19) can be rewritten as

ν = (μ_{1} + β_{1}μ_{s}, …, μ_{s−1} + β_{s−1}μ_{s})ÃB,

where Ã is the (s−1)×r matrix obtained by removing the last row of A. Again ν = 0, but then our induction hypothesis with r replaced by r − 1 allows us to conclude that Ã = 0.

6. Elliptic curves

Let E_{1} and E_{2} be elliptic curves over ℂ. Let G := Ext_{ℂ}^{1}(E_{1}, E_{2}) be the extension group in the category of complex abelian varieties. By the Poincaré reducibility theorem, this is a torsion group. The set of complex points E(ℂ) of an elliptic curve E over ℂ can be viewed as a complex torus written E^{an}; so one may also consider A := Ext^{1}(E_{1}^{an}, E_{2}^{an}), the extension group in the category of complex tori. There is a natural homomorphism G → A; the image is the torsion subgroup of A (see [4, Rmk. 6.2, p. 23]).

We present next a theorem of Lichtenbaum (1960’s, unpublished); however, our proof is different in that it is based on periods. By comparison to Example 5.6 and Proposition 5.7, the theorem below indicates that the situation for elliptic curves runs quite parallel to our own. Specifically, in light of in Example 5.6, it suggests that Ext^{1}(E, F) will rarely be representable as a t-module for nonisogenous Drinfeld modules E and F of rank 2. Moreover, it suggests that Ext^{1}(E, F), for general t-modules E and F, will not always have the structure of a t-module.

Theorem 6.1 (Lichtenbaum). Let E_{1} and E_{2} be elliptic curves over ℂ, and let G := Ext^{1}(E_{1}, E_{2}) and A := Ext^{1}(E_{1}^{an}, E_{2}^{an}).

(a) If E = E_{1} = E_{2} has complex multiplication, then A is naturally isomorphic to E(ℂ) (as abelian groups). Under this isomorphism, the group G is identified with the torsion points of E(ℂ).

(b) If E_{1} and E_{2} are isogenous and admit complex multiplication, then A is isogenous to E_{1}(ℂ) and E_{2}(ℂ).

(c) If at least one of E_{1} and E_{2} does not admit complex multiplication, then the natural topology on G is non-Hausdorff. Therefore, G is not the set of complex points of a complex algebraic variety (with the classical topology).

Proof. Given E_{i} (i = 1, 2) as the quotient of ℂ by lattices ℤ + ℤτ_{i}, where each τ_{i} may be taken to be have positive imaginary part, we see from [4, Prop. 5.7, p. 21] that A is naturally the quotient of ℂ by the subgroup Λ generated by 1, τ_{1}, τ_{2} and τ_{1}τ_{2}.

Let us suppose that an elliptic curve X has complex multiplication; let us think of X(ℂ) as a quotient of ℂ by ℤ + ℤτ. By the theory of complex multiplication [18], we have the following: (i) τ lies in an imaginary quadratic field; (ii) τ^{2} = aτ + b for some integers a and b; and (iii) if X′ (with X′(ℂ) = ℂ/(ℤ + ℤτ′)) is an elliptic curve isogenous to X, then τ′ and τ lie in the same imaginary quadratic field and X′ also has complex multiplication with the same CM-field.

If X(ℂ) = E = E_{1} = E_{2}, then we have τ = τ_{1} = τ_{2}. By (i) and (ii), we get that Λ is the lattice ℤ + ℤτ. Thus we obtain that A is naturally identified with E(ℂ). The last statement in (a) follows from [4, Rmk. 6.2, p. 23].

If E_{1} and E_{2} are isogenous curves with complex multiplication, then τ_{1} and τ_{2} lie in the same imaginary quadratic field F; in this case, Λ is isomorphic to a fractional ideal of an order of K. This proves (b).

For (c), suppose at least one of E_{1} and E_{2} does not admit complex multiplication. By the fundamental theorem of complex multiplication, τ_{1} and τ_{2} are not both contained in an imaginary quadratic field. In other words, in this case Λ is a subgroup of ℤ-rank greater than two, and so it is not a discrete subgroup of ℂ.

Remark 6.2. (C. Schoen) (a) If A and B are abelian varieties defined over \overline{ℚ}, then the natural map Ext1 ; \overline{ℚ}(A, B) → Ext_{ℂ}^{1}(A, B) is an isomorphism.

(b) Any abelian surface corresponding to an element of Ext^{1}(E_{1}, E_{2}) is isogenous to the product E_{1} ×E_{2}. Any complex abelian variety, which is isogenous to a product of CM-elliptic curves, is itself a product of CM-elliptic curves [17].

7. Extensions of t-motives

In this section we explore extensions from the standpoint of t-motives and examine avenues for further study. Given two t-modules E, F over K, we can consider the associated Anderson t-motives M(E) and M(F) [1]. Since the functor M which sends a t-module to its associated t-motive is contravariant, we obtain a map

M^{∗} : Ext^{1}(E, F) → Ext^{1}((M(F), M(E));

because M gives an anti-equivalence of categories of t-modules and t-motives [1, Thm. 1], M^{∗} is an isomorphism. If one is interested in computing just the group of extensions of t-modules, then it is relatively easy to compute extensions in the category of t-motives. We formulate this precisely in the next lemma.

The evident functor f from the category T of t-motives to the category C of left K[t, τ]-modules is fully faithful [1, §1.2]. Here K[t, τ] is the noncommutative ring generated by t and τ with the relations, tτ = τt, xt = tx, τx = x^{q}τ, for all x ∈ K.

Lemma 7.1. For any t-motives A and B over K, the natural map

f^{∗} : Ext_{T}^{1} (A, B) → Ext_{C}^{1}(f(A), f(B))

is an isomorphism.

Proof. The fact that f is fully faithful implies that f^{∗} is injective in the following way. Suppose the images of α and β under f^{∗} coincide. Pick representatives of α and β, i.e. extensions X_{1} and X_{2} of A by B which satisfy f(X_{1}) ≅ f(X_{2}). We obtain a commutative diagram

0 ⟶ f(Y) ⟶ f(X_{1}) ⟶ f(A) ⟶ 0
∥  ^{γ}↓  ∥
0 ⟶ f(Y) ⟶ f(X_{2}) ⟶ f(A) ⟶ 0,

where γ is an isomorphism of K[t, τ]-modules. Since f is fully faithful, γ is an isomorphism of t-motives. More precisely, γ is induced by an isomorphism X_{1} →^{∼} X_{2} of t-motives.

It remains to show the surjectivity of f^{∗}. For this, we have to show the following: given any extension X of f(A) by f(B) in C, the left K[t, τ]-module X is a t-motive, i.e. (i) it is free and finitely generated as a K[τ]-module and (ii) the associated primes of X′ := X/τX, viewed as a module over the commutative ring R := K[t], consist only of the principal ideal I := (t − θ). Geometrically, we want the coherent sheaf associated to X′ on 𝔸^{1} to be supported only at the point t = θ.

Condition (i) is clearly satisfied by X by general properties of modules over the ring K[τ] [12, Prop. 5.4.9]. For (ii), consider an extension Q of P by N where P and N are finitely generated R-modules. Every associated prime of Q is an associated prime of either P or N. Also, if H is a quotient module of N, then every associated prime of H is an associated prime of N (this is easy to see via the geometric interpretation).

Now, by assumption, the associated primes of the R-modules A′ := f(A)/τf(A) and B′ := f(B)/τf(B) consist of just the ideal I. The R-module X′ is an extension of A′ by B′′ (= a quotient module of B′). So we may apply the comments in the previous paragraph to the extension X′ to deduce that X′ satisfies (ii).

Remark 7.2. Lemma 7.1 shows that extensions may be computed via resolutions of t-motives by free K[t, τ]-modules. Furthermore, it implies the injectivity of the map

f^{∗} : Ext_{T}^{2} (A, B) → Ext_{C}^{2}(f(A), f(B)).

For M and N as in Remark 7.4 (and K = 𝔽_{q}), this gives Ext_{T}^{2} (M, N) = 0.

Remark 7.3 (Analogy with D-modules). A. Rosenberg points out that K[t, τ] and, more generally, skew polynomial algebras are analogous to Weyl algebras in D-module theory (the first Weyl algebra is ℂ[x, ∂] with the relation ∂x − x∂ = 1) in that they are all special cases of hyperbolic algebras [16, Ch. II]. Since Ext’s of certain (but not all) D-modules possess a nontrivial structure of a D-module, one expects the same to be true for t-modules.

The analogy with D-modules is best viewed within the context of opers (see [3, §7.3.14]; a Drinfeld module is an example of a Frobenius oper) and noncommutative algebraic geometry (see [5, Rmk. 5.3.5, §6], [7, §0.6], [16]).

Remark 7.4. Geometrically interpreting the definition of a t-motive [1, §1.2], a tmotive M is a special sheaf F_{M} over a noncommutative surface S [16] given by the product of the noncommutative affine line 𝔸_{nc}^{1} by the commutative affine line 𝔸^{1}. We think of S as fibered over 𝔸^{1} (viewed horizontally) with (vertical) fibers 𝔸_{nc}^{1}. The sheaf F_{M} is the ideal sheaf of a curve X_{M} ⊂ S for which the following hold.

(a) X_{M} is finite over the two axes, via the projections to the components; in other words, X_{M} is transversal to the horizontal and vertical fibers.

(b) The intersection of X_{M} with the horizontal axis (which corresponds to τ = 0) is a nilpotent subscheme of 𝔸^{1} supported at the point t = θ; (a) assures us that the intersection is a proper subscheme of 𝔸^{1}.

If M and N are distinct t-motives, then the group Ext_{T}^{1} (M, N) can be interpreted via the “intersection scheme” of the curves X_{M} and X_{N} in S. We can view the subgroup Ext_{0}^{1}(M, N) (cf. Corollary 2.3) of Ext_{T}^{1} (M, N) as the nontrivial part of the intersection locus corresponding to points distinct from t = θ, τ = 0, as in (b).

The geometric situation is especially clear in the case K = 𝔽_{q}: S is the usual commutative affine plane 𝔸^{2}; if g(t, τ) and h(t, τ) (assumed to have no common factors) are defining equations for the curves X_{M} and X_{N}, then Ext_{T}^{1} (M, N) is isomorphic to the quotient module 𝔽_{q}[t, τ]/(g, h) of the commutative ring 𝔽_{q}[t, τ].

Remark 7.5. The results and ideas in this paper are used in an ongoing project with Thakur, whose aim is to relate extension groups of t-modules to values of zeta functions in the spirit of [2], [10], [15].

Acknowledgements

We wish to thank G. Anderson, W. D. Brownawell, D. Goss, S. Lichtenbaum, and D. Thakur for encouragement, useful discussions, and helpful advice. Part of the work on this paper was done when the second author was visiting MPIM (Bonn), which he thanks for its hospitality, support, and excellent working conditions.

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Department of Mathematics, Brown University, Providence, RI 02912 USA

  E-mail address: map@math.brown.edu

Department of Mathematics, University of Maryland, College Park, MD 20742 USA

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: atma@math.umd.edu