Descent for differential modules and skew fields

Mark van Hoeij^{∗} and Marius van der Put^{∗∗}

∗Department of Mathematics, Florida State University Tallahassee, FL 32306, USA. email: hoeij@math.fsu.edu

∗∗Department of Mathematics, University of Groningen P.O.Box 800, 9700 AV Groningen, The Netherlands email: mvdput@math.rug.nl

Abstract. In this paper we study rationality questions for differential modules and differential operators. If a differential operator L is equivalent to its conjugates over k, is it then equivalent to an operator defined over k? We will show how counter examples to this question correspond to skew fields, and we will make this correspondence explicit in both directions. Similar questions are studied for projective equivalence of differential operators. The main tool is the study of differential modules over skew fields.

1 Introduction

1.1 Examples of descent phenomena

Let K/k be a Galois extension of differential fields of characteristic 0. The skew ring of differential operators over K is denoted by D := K[∂]. An element σ in the Galois group Gal(K/k) acts on D by σ(∑ a_{i}∂^{i}) = ∑ σ(a_{i})∂^{i}. The conjugates of an element L ∈ D are the operators σ(L). The order of

∗Supported by NSF grant 0098034.

L is the degree in ∂. If L ∈ k[∂] then L is called rational. Two operators L_{1}, L_{2} are called equivalent (or of the same type) if the two differential modules D/DL_{i}, i = 1, 2 are isomorphic. The operators are called projectively equivalent if there exists a D-module E of dimension 1 over K such that D/DL_{1} ≅ E ⊗ D/DL_{2}.

Abbreviations:

R Rational, i.e. element of k[∂].
E_{R} Equivalent to an element of k[∂].
E_{C} Equivalent to all its conjugates over k.
E_{R}^{p} Projectively equivalent to an element of k[∂].
E_{C}^{p} Projectively equivalent all its conjugates over k.

The following implications are obvious:

R ⟹ E_{R}⟹ (E_{C} and E_{R}^{p}), (E_{C} or E_{R}^{p}) ⟹ E_{C}^{p}.

These implications leave seven possible combinations for the truth-values of R,E_{R},E_{C},E_{R}^{p},E_{C}^{p}. In case k = Q(x) and K = \overline{Q}(x), all seven cases occur. The examples, given in the table below, are irreducible in D.

Case R E_{R} E_{C} E_{R}^{p} E_{C}^{p} Example
1 Y Y Y Y Y ∂
2 N Y Y Y Y ∂ − 1/(x − i)
3 N N Y Y Y (x^{2} + 1)∂^{2} + (2x + i)∂ + 1
4 N N Y N Y ∂^{2} + x^{2} + 1 + i
5 N N N Y Y ∂ + i
6 N N N N Y 4x∂^{2} − 2∂ + x^{2} + x + i
7 N N N N N ∂^{2} + i

Cases 1 and 7 are not of interest. Further, [H] p. 101/102 contains already an example of a second order operator R_{s,t} for which computer computations show that R_{s,t} is E_{C} but not E_{R}. The table can be verified by computer computations, however, this would not explain the underlying mathematics nor where these examples come from. That is the main theme of this paper.

Denote C_{K} resp. C_{k} as the field constants of K resp. k. Suppose that K = C_{K}(x) and k = C_{k}(x) with differentiation \frac{d}{dx}. Then the descent phenomenum “E_{C} without E_{R}” (case 3 or 4) is related to skew fields F^{o} of finite dimension over their center C_{k}. The main result in section 2 is that any such skew field yields examples for “E_{C} without E_{R}”. Case 6 (a descent phenomenum up to projective equivalence) also corresponds to a skew field F, this time of finite dimension over its center k. For second order equations this skew field is a quaternion field and corresponds to a conic over k. The main result in section 3 is that all skew fields of this type (all non-degenerate conics) produce examples for case 6. For most of our constructions it will be more convenient to use modules instead of operators. We used operators in the above table for compactness of notation.

1.2 Descent problems

In section 2, one considers a differential module M over K which is isomorphic to all its conjugates ^{σ}M under the Galois group of K/k. The question is whether M descends to k, i.e., M ≅ K ⊗_{k} N for some differential module N over k. The obstruction to descent is a 2-cocycle which corresponds to a skew field, see theorem 2.10 and proposition 2.11 for a general differential field k, with additional results for k = C_{k}((x)) in theorem 2.4 and for k = C_{k}(x) in theorem 2.8. In section 2.4 we show how all possible examples over k = C_{k}(x) can be constructed. For completeness, Amitsur’s completely different construction (which makes a very special case of the descent problem explicit, namely part (a) of proposition 2.11) is presented in section 2.5.

In section 3 we study the problem whether a 3-dimensional differential module M is the second symmetric power of a 2-dimensional differential module N. At the heart of the rationality issues in this problem is a conic, and we show that every conic over k = C_{k}(x) occurs. A surprising result is theorem 4.7 in section 4, which states that the isomorphism class of a solution N of this problem is unique up to tensoring with 1-dimensional modules, if the field of constants is algebraically closed. This implies that the rationality issues in section 3 can also be viewed as a “projective descent problem”, which is the subject of theorem 4.4. In this type of descent problem we consider modules that are not necessarily isomorphic to their conjugates, but only isomorphic up to tensoring with 1-dimensional modules.

1.3 Motivation and overview of the main results

In section 2.3 we follow standard Galois cohomology techniques to describe descent phenomena for differential modules. This way one can explain the descent phenomena in terms of 2-cocycles or skew fields, and classify descent phenomena in one direction “a differential module that is isomorphic to its conjugates ⟹ a skew field” but not in the opposite direction: Which skew fields occur this way? Our goal is a thorough classification and hence we study this question in detail for several common differential fields. In theorem 2.4 in section 2.1 we show that over the field of formal Laurent series only the trivial case occurs (there is no obstruction to descent) whereas in section 2.4 we show that over the rational functions every skew field occurs. Each of these results require completely different techniques. The key ingredient in our construction “skew field ⟹ differential module” is the introduction of differential modules over skew fields, which are then viewed as differential modules over a commutative subfield. To illustrate this idea we give explicit examples. We then give an construction that we prove to be complete (it provides explicit examples for every skew field, and every example can be constructed this way). Our construction also implies an interesting complexity result, namely that factoring fourth order operators in Q(x)[∂] is at least as hard as finding rational points on a conic over Q, a problem related to factoring integers, which is generally assumed to be very hard.

One of the motivations to study the descent problem is the question which extensions of the constants need to be considered in algorithms for solving differential equations, and the results in section 2 are of interest for factoring differential operators.

Section 3 is motivated by the problem of trying to reduce a third order linear differential equation to a second order equation by determining if the corresponding 3-dimensional differential module is the symmetric square of a 2-dimensional module. This problem requires finding a point on a conic that is defined over the differential field k. Such a conic corresponds to a skew field of dimension 4 over its center k. A natural question is: Can every conic occur? For the formal Laurent series again only the trivial case occurs, but for the rational functions k = C_{k}(x) we show explicitly in section 3.2 that every conic over k occurs. This result implies that if an algorithm tries to reduce a third order differential equation by trying to write the corresponding module as a symmetric square, then a high degree extension of the constants could be necessary, see section 3.2.2. Our construction in section 3 differs from the one in section 2, it uses skew fields over k instead of over C_{k} (the field of constants of k).

In theorem 4.7 in section 4 we show that if a 3-dimensional differential module is the symmetric square of a 2-dimensional module, then this 2dimensional module is unique up to projective equivalence, provided that the field of constants is algebraically closed. Corollary 4.2 shows that this is then also true over C_{k}(x) without C_{k} algebraically closed except in the imprimitive case, and to complete the result we give counter examples for the imprimitive case in examples 4.3. The conic occurs in sections 3 and 4 as 1-dimensional submodule of the symmetric square of a 3-dimensional module. How to view the skew field that corresponds to this conic in terms of Galois cohomology is shown in theorem 4.4 combined with 4.7.

We would like to thank J.-A. Weil and E. Compoint for sharing their ideas in [C-W].

2 The Descent problem

2.1 Twists and descent data

Let K ⊃ k denote two fields and let M be some object over K. The descent problem asks for an object N over k such that K ⊗_{k} N is isomorphic to M. We are interested in the case where M and N are differential modules. We start with definitions and notations.

Definitions 2.1 The twist ^{σ}V of a vector space V. Let K be a field and σ an automorphism of K. For any vector space V over K one associates a vector space ^{σ}V which is equal to V as additive group and has a new scalar multiplication defined by λ ∗ v := σ^{−1}(λ)v for all λ ∈ K and v ∈ V. One has ^{σ_{1}} (^{σ_{2}}V) = ^{σ_{1}σ_{2}}V. For any K-linear map f : V → W between K-vector spaces one denotes by ^{σ}f : ^{σ}V → ^{σ}W the same map f, which is K-linear for the new structures. In this way, one has defined a functor from the category of the K-vector spaces to itself. This functor commutes with tensor products, exterior powers, symmetric powers, et cetera. If b is the matrix of f with respect to a basis of V and a basis of W, then σ(b) is the matrix of ^{σ}f with respect to the same bases. Here σ acts on b by acting on the entries.

Let V, W be K-vector spaces and let σ be an automorphism of K. The map f : V → W is called σ-linear if f is additive and f(λv) = σ(λ)f(v) for all λ ∈ K, v ∈ V. To a linear map f : ^{σ}V → W we associate a σ-linear map F : V → W by composing the set-theoretic identity V → ^{σ}V with f.

Observation: Let K ⊃ k be a finite Galois extension with Galois group G. Let V be a vector space over K. Then there exists a natural linear map from K ⊗_{k} V to ^{σ}V, and a natural isomorphism from K ⊗_{k} V to ⊕_{σ∈G} ^{σ}V.

The maps are given by a ⊗ v ↦ a ∗ v = σ^{−1}(a)v. The map to ⊕_{σ∈G} ^{σ}V is one to one (hence onto by comparing dimensions) because of the linear independence of automorphisms.

Definitions 2.2 Descent data and the descent problem.

(1) k is a differential field of characteristic zero. Its algebraic closure will be denoted by \overline{k}. Let K ⊂ \overline{k} be a Galois extension (finite or infinite) of k. Then K is also a differential field. The action of the Galois group G of K/k commutes with differentiation on K. Let D := K[∂] denote the skew ring of the differential operators over the field K. A differential module over K is a left D-module, of finite dimension as vector space over K. Let M be a differential module over K. For σ ∈ G one defines the twist ^{σ}M of M as follows: The D-module ^{σ}M is M as additive group, has a new scalar multiplication as defined in 2.1 and has the same operator ∂.

The action of G on K is extended to an action on D by imposing σ(∂) = ∂ for all σ ∈ G. As usual, one associates to a monic differential operator L ∈ D the differential module M = D/DL. For σ ∈ G one has that ^{σ}M is the differential module associated to σ(L).

(2) Descent data for M are given by:

(i) for each σ ∈ G an isomorphism ϕ(σ) : ^{σ}M → M of differential modules, (ii) satisfying ϕ(σ) ^{σ}ϕ(τ) = ϕ(στ) for all σ, τ ∈ G.

Let Φ(σ) : M → M be the σ-linear map associated to ϕ(σ). Then the two conditions can be formulated as follows:

(i) Φ(σ) : M → M is a σ-linear bijection, commuting with ∂, (ii) satisfying Φ(σ)Φ(τ) = Φ(στ) for all σ, τ ∈ G.

The above definition coincides with the one used in algebraic geometry. Indeed, let K/k be a finite Galois extension. The algebra K ⊗_{k} K has a left and a right K-algebra structure. Let N_{l} := (K ⊗_{k} K) ⊗_{K} M and N_{r} := (K ⊗_{k} K)⊗_{K} M denote the two tensor products w.r.t. the left and the right structure. The first part (I) of the descent data is a K ⊗_{k} K-linear isomorphism N_{l} → N_{r} which commutes with ∂. The second part (II) of the descent data is a relation between the various structures on (K⊗_{k}K⊗_{k}K)⊗_{K}M. Now N_{l} is isomorphic with the direct sum of [K : k] copies of M and N_{r} ≅ K ⊗_{k} M is isomorphic to ⊕_{σ∈G} ^{σ}M. Therefore (I) is equivalent to (i). Furthermore, one can show that (II) is equivalent with (ii).

(3) One says that M descends to k if there exists a differential module N over k such that M is isomorphic to K ⊗_{k} N. If M descends to k, then M has obviously descent data. On the other hand, let descent data for M be given. Choose a basis m_{1}, .., m_{n} of M over K. This also provides a basis for ^{σ}M using the set-theoretic identity. Let b(σ) denote the matrix of ϕ(σ) w.r.t. this basis. Then condition (ii) translates into: (ii) b(στ) = b(σ) σ(b(τ)) in other words σ ↦ b(σ) is a 1-cocycle for G and GL_{n}(K). Here σ ∈ G acts on a matrix by acting on its entries. It is well known that H^{1}(G, GL_{n}(K)) is trivial (see [S]) and that this implies that there exists a K-basis m̃_{1}, …, m̃_{n} of M such that the matrices b̃(σ) of ϕ(σ) w.r.t. this new basis are 1. Now N := km̃_{1} ⊕ ⋯ ⊕ km̃_{n} = {m ∈ M|∀_{σ} Φ(σ)(m) = m} is a differential module over k because the Φ(σ) commute with ∂. The natural map K ⊗_{k} N → M is an isomorphism of differential modules, so M descends to k.

(4) In the sequel we will study the situation where only the first part of the descent data is given, i.e., for a differential module M over K a collection of isomorphisms {ϕ(σ) : ^{σ}M → M} is given. The descent problem is to decide whether M descends to k or, more generally, to find the differential fields ℓ (say with K ⊃ ℓ ⊃ k) such that M descends to ℓ. This problem can be stated for differential operators as follows: Assume that L is E_{C} (see abbreviations in section 1), is L also E_{R}?

(5) For any algebraic extension of differential fields K ⊃ k one can also define descent data and a weak form of descent data, as in (4), for a differential module over K. Let K_{c} denote the normal closure of K. Then these data translate into data for N := K_{c} ⊗_{K} M w.r.t. the Galois extension K_{c}/k and with the additional information that N descends to K. Therefore we will restrict ourselves to Galois extensions.

We start with a useful lemma. Consider an algebraic extension of differential fields k ⊂ K, obtained by extension of constants, and a differential module M over K which descends to k. Then, by part (a) of the lemma, the module N over k with K ⊗_{k} N ≅ M is unique up to isomorphism.

Lemma 2.3 Consider an algebraic extension of differential fields k ⊂ K, with fields of constants C_{k}, C_{K}. Suppose that K = C_{K} · k.

(a) Let M_{1}, M_{2} be differential modules over K, which descend to modules N_{1}, N_{2} over k. Then

Hom_{K[∂]}(M_{1}, M_{2}) = C_{K} ⊗_{C_{k}} Hom_{k[∂]}(N_{1}, N_{2}).

Furthermore, if M_{1} ≅ M_{2}, then N_{1} ≅ N_{2}.

(b) Let N be a differential module over k and put M = K ⊗_{k} N. If the group Aut_{k[∂]}(N) of differential automorphisms of N is C_{k}^{∗}, then End_{k[∂]}(N) = C_{k}, and End_{K[∂]}(M) = C_{K}. Moreover, Aut_{K[∂]}(M) = C_{K}^{∗}.

Proof. (a) Let W := Hom_{k[∂]}(N_{1}, N_{2}) denote the C_{k}-vector space of the differential homomorphisms between the two differential modules over k. The natural C_{K}-linear map C_{K} ⊗ W → Hom_{K[∂]}(M_{1}, M_{2}) is a bijection. Indeed, let 1 denote the trivial differential module of dimension one over k. Then Hom_{k[∂]}(N_{1}, N_{2}) ≅ Hom_{k[∂]}(1, N_{1}^{∗} ⊗ N_{2}) and the latter is the space {a ∈ N_{1}^{∗} ⊗ N_{2}| ∂a = 0} of the “rational solutions” of the differential module N_{1}^{∗} ⊗N_{2}. Similarly, Hom_{K[∂]}(M_{1}, M_{2}) is the space of the rational solutions of M_{1}^{∗}⊗M_{2} = K⊗_{k}(N_{1}^{∗}⊗N_{2}). The bijectivity follows now from [P-S], chapter 4, proposition 4.3. Consider a basis w_{1}, …, w_{d} of W. There is a polynomial h ∈ K[X_{1}, …, X_{d}] (the determinant of ∑ X_{i}w_{i} w.r.t. any bases for N_{1}, N_{2}) such that the set {(λ_{1}, …, λ_{d}) ∈ C_{K}^{d}| ∑_{i} λ_{i}w_{i} is an isomorphism} is given by h(λ_{1}, …, λ_{d}) ≠ 0. This implies that some w ∈ W induces an isomorphism between M_{1} and M_{2}. Then w is an isomorphism between N_{1} and N_{2}.

(b) Put E := End_{k[∂]}(N) = Hom_{k[∂]}(N, N). This is a finite dimensional algebra over C_{k} with basis e_{1}, …, e_{d}. For any e = ∑ λ_{i}e_{i} one writes det(e) for the determinant of the matrix of left multiplication by e on N. This is a polynomial in λ_{1}, …, λ_{d}. The automorphisms of N are the elements e ∈ E with det(e) ≠ 0. The assumption that this group is C_{k}^{∗} implies that E = C_{k}. Then, as in (a), End_{K[∂]}(M) = C_{K} and Aut_{K[∂]}(M) = C_{K}^{∗}. □

Remark: If K ≠ C_{K} · k then the the lemma does not hold in general. Let M = K ⊗_{k} N_{1} and assume Aut_{K[∂]}(M) = C_{K}^{∗}. Then M descends to a differential module N_{2} over k if and only if there exists a one-dimensional differential module L over k for which K ⊗_{k} L is trivial and N_{2} ≅ L ⊗_{k} N_{1}. See also lemma 2.7.1 in [Ka].

Theorem 2.4 Let C_{K}/C_{k} be a finite Galois extension with group G. Consider the differential fields k = C_{k}((x)) and K = C_{K}((x)) with differentiation x\frac{d}{dx}. Suppose that the differential module M over K has the property: ^{σ}M is isomorphic to M for all σ ∈ G. Then M descends to k.

Proof. First we treat the case that M is irreducible. The proof is based on the classification of irreducible differential modules over K as given in the thesis [So] of R. Sommeling. The relevant information is the following:

(1) One associates to Q ∈ ⋃_{m≥1} \overline{C_{K}}[x^{−1/m}] the one dimensional differential module K(Q)e over the finite field extension K(Q) of K, given by ∂e = Qe (recall that ∂ now refers to x\frac{d}{dx} instead of \frac{d}{dx}). Then K(Q)e, viewed as a differential module over K, will be denoted as E(Q). It is irreducible and has dimension [K(Q) : K].

(2) Every irreducible differential module over K is obtained in this way.

(3) E(Q_{1}) ≅ E(Q_{2}) if and only if there exists a τ in the Galois group of \overline{K}/K such that τ(Q_{1}) − Q_{2} ∈ \frac{1}{r}Z, where r is the ramification index of K(Q_{1})/K (or equivalently the ramification index of K(Q_{2})/K).

One associates to Q (as above) its monic minimal polynomial f_{Q} ∈ K[T] over K. Then (3) translates into ([So], Proposition 3.3.6): (3’) E(Q_{1}) ≅ E(Q_{2}) if and only if there is a λ ∈ \frac{1}{r}Z, where r is the ramification index of K(Q_{1})/K, such that f_{Q_{1}}(T) = f_{Q_{2}}(T + λ).

Now suppose that ^{σ}M is isomorphic to M for all σ. Choose a Q ∈ \overline{C_{K}}[x^{−1/r}] with r ≥ 1 minimal, such that M ≅ K(Q)e. Let f_{M} = f_{Q} ∈ K[T] be the minimal monic polynomial of Q over K. Take σ ∈ Gal(C_{K}/C_{k}) = Gal(K/k) and extend σ to an automorphism σ̃ of \overline{K}/k. Then ^{σ}M is seen to be associated to σ̃(Q). Therefore f_{^{σ}M} = σ(f_{M}). Thus ^{σ}M ≅ M translates into σ(f_{M})(T) = f_{M}(T + λ) for some λ ∈ \frac{1}{r}Z. Since σ has finite order, one has λ = 0. Hence f_{M} = f_{Q} ∈ k[T]. Now Q defines also an irreducible differential module N := k(Q)e with ∂e = Qe over k of dimension [k(Q) : k]. Clearly K ⊗_{k} N is isomorphic to M. This completes the proof in case M is irreducible.

We now sketch the proof that the assumption that M is irreducible can be omitted. Any differential module M can be written as a finite direct sum ⊕_{i}E(Q_{i}) ⊗ R_{i}, where E(Q_{i}) ≅ E̸(Q_{j}) for i ≠ j and where each R_{i} has the property that the matrix of ∂ w.r.t. a basis is nilpotent. This decomposition is unique (see [L]) and the Q_{i} are unique up to the equivalence stated in the proof of 2.4. One observes that each R_{i} descends to k. This information suffices to prove the theorem without the assumption that M is irreducible. □

Remarks 2.5

(1) The following example illustrates that theorem 2.4 does not hold for positive characteristic. Let 𝔽_{2} resp. 𝔽_{4} = 𝔽_{2}(α) denote the fields with 2 resp. 4 elements. Let σ be the non-trivial automorphism σ(α) = α^{2} = α + 1. Let k = 𝔽_{2}((x)) and K = 𝔽_{4}((x)) with differentiation x′ = 1. Let M = Ke with ∂e = \frac{α}{x}e. Then e ↦ x e defines an isomorphism ^{σ}M ≅ M, but M does not descend to k. In the rest of this paper we will only consider characteristic 0.

(2) Let K be any finite (Galois) extension of C_{k}((x)). Such K has the form C_{K}((t)) where t has the property t^{m} = cx with c ∈ C_{K}^{∗}. The above theorem and its proof remain valid for this K and any differential module M over K. However, in case m > 1, the differential module N over k with K ⊗_{k} N ≅ M is no longer unique up to isomorphism.

(3) Theorem 2.4 and (1), (2) above, answer the descent problem for formal Laurent series fields. For convergent Laurent series field, the situation is quite different, see [P]. In the sequel of this paper we will study the descent problem for the global case, i.e., for differential fields which are function fields in one variable over the field of constants. The situation k = C_{k}(x) ⊂ K = C_{K}(x), where C_{k} ⊂ C_{K} is an algebraic extension, is easier to deal with (see theorem 2.8) than the general case. The following example illustrates this.

(4) Consider the differential fields k = Q(x) and K = Q(t) with x = t − \frac{1}{t} and with differentiation ′ given by x′ = 1. Let σ denote the non-trivial automorphism of K/k. We note that σt = −\frac{1}{t}. Define the 1-dimensional differential module M = Ke with ∂e = \frac{t′}{2t}e. Then ^{σ}M ≅ Ke with ∂e = −\frac{t′}{2t}e and thus M is isomorphic to ^{σ}M. We will prove that there does not exist h ∈ K^{∗} such that a := \frac{t′}{2t} + \frac{h′}{h} ∈ k. This implies that M does not descend to k. Suppose that h exists. Then σa = a and consequently \frac{t′}{t} = \frac{σh′}{σh} − \frac{h′}{h}. Then t = α\frac{σh}{h} for some α ∈ Q. This yields the contradiction −1 = σ(t)t = α^{2}.

2.2 Semi-simple modules and semi-simple algebras

In this subsection some known facts are collected that are useful for the descent problem. It seems that proposition 2.7 is not available is the literature. For the first standard result we omit the proof.

Lemma 2.6 Let M be a differential module over K. The following are equivalent.

(1) M is a sum of irreducible submodules.

(2) M is a direct sum of irreducible submodules.

(3) Every submodule N ⊂ M is a direct summand, i.e., there exists a submodule N′ with M = N ⊕ N′.

A differential module M, having the equivalent properties of 2.6, is called semi-simple or completely reducible.

Proposition 2.7 Let K ⊃ k be an algebraic extension of differential fields.

(1) Let N be a differential module over k. Then N is semi-simple ⇔ the differential module K ⊗_{k} N over K is semi-simple.

(2) Suppose that [K : k] < ∞ and that M is a semi-simple differential module over K. Then M considered as a differential module over k is semi-simple.

(3) Let A, B be semi-simple differential modules over k. Then A ⊗_{k} B is semi-simple.

(4) The collection of semi-simple differential modules is stable under all “operations of linear algebra”, i.e., submodules, quotients, direct sums, tensor products, symmetric powers,...

Proof. (1’). We suppose first that K ⊃ k is a finite Galois extension with Galois group G. (1’)⇒. We may suppose that N is irreducible. Take an irreducible Ksubmodule D of K ⊗_{k} N. Every σ ∈ G acts on K ⊗_{k} N in the obvious way and this action commutes with ∂. Then each σ(D) is also an irreducible K-submodule of K ⊗_{k} N. Therefore E = ∑_{σ∈G} σ(D) is a semi-simple Ksubmodule. Moreover, σ(E) = E for all σ ∈ G. Hence E = K ⊗_{k} F for some submodule of N. Since N is irreducible, one has that F = N and E = K ⊗_{k} N. (1’)⇐. Let F ⊂ N be a submodule. Then K ⊗_{k} F is a K-submodule of K ⊗_{k} N. There exists a K-linear P : K ⊗_{k} N → K ⊗_{k} N with the properties im P = K ⊗_{k} F, P^{2} = P and P ∂ = ∂P. Define Q = \frac{1}{|G|} ∑_{σ∈G} σP σ^{−1}. Then Q∂ = ∂Q, Qm = m for m ∈ K ⊗_{k} F, im Q = K ⊗_{k} F and τQ = Qτ for all τ ∈ G. Then Q^{2} = Q. Let us identify any n ∈ N with 1 ⊗ n ∈ K ⊗_{k} N. For any n ∈ N one has that Q(n) is invariant under G and therefore belongs to N. The restriction Q̃ of Q to N has the properties im Q̃ = F, Q̃∂ = ∂Q̃ and Q̃^{2} = Q̃. Then N = F ⊕ ker Q̃. Thus N is semi-simple.

(1”). Consider any finite extension of differential fields k ⊂ K. Let K_{c} denote its normal closure. By (1’), N is semi-simple over k if and only if K_{c}⊗_{k}N is semi-simple over K_{c}. But, again by (1’), K_{c}⊗_{k}N = K_{c}⊗_{K}(K⊗_{k}N) is semi-simple over K_{c} if and only if K ⊗_{k} N is semi-simple over K.

(1). Let K ⊃ k be an arbitrary algebraic extension. Suppose that N is semi-simple. Any K-submodule D of K ⊗_{k} N comes (by tensoring) from a K̃-submodule D̃ of K̃ ⊗_{k} N for some field K̃ ⊂ K which is finite over k. Since D̃ is a direct summand, D is a direct summand, too. Suppose that K ⊗_{k} N is semi-simple. Let F ⊂ N be a submodule. Then K ⊗_{k} F is a direct summand of K ⊗_{k} N. For a suitable finite extension K_{1} of k, contained in K, also K_{1} ⊗_{k} F is a direct summand of K_{1} ⊗_{k} N. One replaces K_{1} by its normal closure K_{2} ⊃ K_{1}. Then K_{2} ⊗_{k} F is a direct summand of K_{2} ⊗_{k} N. As in the proof of (1’)⇐, it follows that F is a direct summand.

(2). Let K_{c} be the normal closure of K. Then K_{c} ⊗_{K} M is semi-simple over K_{c}. Further M is a k-submodule of K_{c} ⊗_{K} M. Therefore it suffices to consider the case where K is a Galois extension of k with Galois group G. As in 2.1 one has K ⊗_{k} M ≅ ⊕ ^{σ}M. Each ^{σ}M is semi-simple and thus K ⊗_{k} M is a semi-simple differential module over K. By (1), M is semi-simple as a k-differential module.

(3) We use the notations: C_{k} is the field of constants of k, \overline{C}_{k} is an algebraic closure of C_{k} and \underline{k} = \overline{C}_{k} · k. By (1), \underline{A} := \underline{k} ⊗_{k} A and \underline{B} := \underline{k} ⊗_{k} B are semi-simple. Using differential Galois theory, see [P-S], Exercise 2.38 (4), one has that \underline{A} ⊗\underline{k} \underline{B} is semi-simple. By (1), A ⊗_{k} B is semi-simple.

(4) follows from (3). □

Let M be a semi-simple differential module over k. The isotypical decomposition M = M_{1} ⊕ ⋯ ⊕ M_{s} is defined by: each M_{i} is a direct sum of, say, n_{i} copies of an irreducible differential module D_{i}. Moreover the D_{i} ≢ D_{j} for i ≠ j. Put F_{i} = End_{k[∂]}(D_{i}). Then one has the following obvious results:

(a) The isotypical decomposition of M is unique.

(b) F_{i} is a (skew) field of finite dimension over the field of constants C_{k} of k.

(c) End_{k[∂]}(M) is isomorphic to the product of the algebras Matr(n_{i}, F_{i}).

Here Matr(n, F) denotes the ring of n by n matrices with entries in F. Let C be any field. The algebras A that we consider here are supposed to have a neutral element, further C lies in the center of A, and A as vector space over C has finite dimension. A is called semi-simple if for every twosided ideal I, there is a two-sided ideal J with A = I ⊕ J. The semi-simple algebras are the algebras of the form ∏_{i} Matr(n_{i}, F_{i}) where the F_{i} are (skew) fields of finite dimension over C, with C in the center of F_{i}. The algebra A is called simple if A has no two-sided ideals other than 0 and A. Equivalently, A ≅ Matr(n, F) for some n and some (skew) field F of finite dimension over C, with C in the center of F.

In the remainder of this section we recall some standard facts on skew fields and the Brauer group. For more information we refer to [Bl,Bo,Rei,S]. Let C be any field. Consider a skew field F of finite dimension over its center C. Then the dimension of F over C is a square, say n^{2}. A field extension C′ ⊃ C is called a splitting field for F if C′ ⊗ F is isomorphic to the matrix algebra Matr(n, C′). Any maximal commutative subfield C′ of F satisfies [C′ : C] = n and is moreover a splitting field for F of minimal degree over C. This is illustrated by the example of Hamilton’s quaternion field over Q, namely H = Q + Qi + Qj + Qk. The minimal splitting fields are Q(\sqrt{−m}) for every squarefree positive integer m that can be written as the sum of three squares in Q.

A central simple algebra A over the field C is an algebra whose only twosided ideals are A and {0} and which has finite dimension over its center C. Every central simple algebra over C has the form Matr(d, F), where F is a (skew) field with center C. The Brauer group Br(C) of a field C consists of the equivalence classes [A] of the central simple algebras over C. Two such algebras A_{i} = Matr(d_{i}, F_{i}) are called equivalent, [A_{1}] = [A_{2}], if the (skew) fields F_{1}, F_{2} are isomorphic. The group structure on Br(C) is induced by the tensor product.

If C_{K}/C_{k} is a finite Galois extension with group G then one defines the following subgroup of Br(C_{k})

Br(C_{K}/C_{k}) = {[A] ∈ Br(C_{k}) | A is split by C_{K}}.

Let c be a 2-cocycle with values in C_{K}^{∗}, i.e. c(σ, τ) ∈ C_{K}^{∗} for all σ, τ ∈ G and c satisfies the 2-cocycle relation. Now c is called normalized if c(1, σ) = c(σ, 1) = 1 for all σ. The image of c in H^{2}(G, C_{K}^{∗}) is trivial, i.e. \overline{c} = 1, if and only if there exists a map f : G → C_{K}^{∗} such that c(σ, τ)f(στ) = f(σ)σ(f(τ)) for all σ, τ. Now take the C_{K}-vector space A with basis {b_{σ} | σ ∈ G} where b_{1} = 1. One turns A into an algebra (a so-called crossed-product algebra) with multiplication rules: b_{σ} · b_{τ} = c(σ, τ)b_{στ}, and b_{σ} · λ = σ(λ)b_{σ} for λ ∈ C_{K}. The 2-cocycle relation guarantees that A is associative, in fact, A is a central simple algebra over C_{k}. Now \overline{c} ↦ [A] defines an isomorphism

H^{2}(G, C_{K}^{∗}) ≅ Br(C_{K}/C_{k}).

Taking the limit over all finite Galois extensions one finds

H^{2}(Gal(\overline{C_{k}}/C_{k}), \overline{C_{k}}^{∗}) ≅ Br(C_{k}).

The inflation homomorphism H^{2}(G, C_{K}^{∗}) → H^{2}(Gal(\overline{C_{k}}/C_{k}), \overline{C_{k}}^{∗}) is injective, and corresponds to the embedding of Br(C_{K}/C_{k}) in Br(C_{k}).

2.3 The associated two-cocycle and skew fields

Theorem 2.8 The associated 2-cocycle for a special case. Consider the differential fields k = C_{k}(x) and K = C_{K}(x) where C_{K}/C_{k} is a finite Galois extension with group G. Assume that the differential module M of dimension n over K has the properties:

(i) The group of the differential automorphism of M is C_{K}^{∗}. (ii) For every σ ∈ G one has ^{σ}M ≅ M. These assumptions define a 2-cocycle class \overline{c} ∈ H^{2}(G, C_{K}^{∗}) which has the following properties:

(a) \overline{c} = 1 ⇔ M descends to k.

(b) The order d of \overline{c} divides n and [K : k].

(c) \overline{c} ∈ H^{2}(G, C_{K}^{∗}) ≅ Br(C_{K}/C_{k}) ⊂ Br(C_{k}) determines a (skew) field F^{o} with center C_{k}.

(d) E := End_{k[∂]}(M) is a simple algebra, of dimension [C_{K} : C_{k}]^{2} over its center C_{k}, having image \overline{c} ∈ Br(C_{k}). In particular, E is isomorphic to Matr(m, F^{o}) for some m ≥ 1. Moreover, C_{K} is a maximal commutative subfield of E.

(e) A finite field extension ℓ ⊃ C_{k} is a splitting field for F^{o} if and only if M descends to ℓ(x).

Proof. Choose an isomorphism ϕ(σ) : ^{σ}M → M for each σ ∈ G. By assumption (i), ϕ(σ) is unique up to an element in C_{K}^{∗}. Let Φ(σ) : M → M denote the associated σ-linear map. For σ, τ ∈ G one defines c(σ, τ) := Φ(σ)Φ(τ)Φ(στ)^{−1}. This is a K-linear bijection on M and commutes with ∂. By assumption (i), c(σ, τ) ∈ C_{K}^{∗}. Further c(σ, τ) satisfies the usual 2-cocycle relation. The image \overline{c} ∈ H^{2}(G, C_{K}^{∗}) is independent of the choices for the {ϕ(σ)}. Part (a) follows at once from definitions 2.2 part (3).

(b) Consider the case where M has dimension 1. Write M = Ke and write ∂e = ae. For any σ one normalizes Φ(σ) : M → M by Φ(σ)e = b(σ)e such that b(σ) ∈ C_{K}(x) has the form b_{1}(σ)/b_{2}(σ) with b_{i}(σ) monic elements of C_{K}[x]. In other words, the first coefficient in the series expansion of b(σ) at x = ∞ is 1. Then c(σ, τ) ∈ C_{K}^{∗} has this form too, which implies c(σ, τ) = 1.

Let M of dimension n induce the 2-cocycle class \overline{c}. Then the 2-cocycle class of the 1-dimensional differential modules Λ^{n}M is easily seen to be \overline{c}^{n}. Since the latter is trivial, the order d of \overline{c} is a divisor of n. Moreover, every element in H^{2}(G, C_{K}^{∗}) is annihilated by the order of G, which is [K : k]. This proves (b). The statement in (c) is a standard fact on Brauer groups, see the previous section.

(d) According to 2.3 (a), End_{K[∂]}(K ⊗_{k} M) ≅ C_{K} ⊗_{C_{k}} End_{k[∂]}(M). By the observation in 2.1, K ⊗_{k} M ≅ ⊕_{σ∈G} ^{σ}M, which by assumption (ii) is isomorphic to the direct sum of [C_{K} : C_{k}] copies M. Now End_{K[∂]}(M) = C_{K}, so End_{K[∂]}(K ⊗_{k} M) is isomorphic to the matrix algebra Matr([C_{K} : C_{k}], C_{K}). It follows that the dimension of E over C_{k} is [C_{K} : C_{k}]^{2}.

Consider the algebra A ⊂ E consisting of the L ∈ E of the form L = ∑_{σ∈G} c_{σ}Φ(σ) with c_{σ} ∈ C_{K}. It is easily verified that L = 0 if and only all c_{σ} are 0. A is a crossed-product algebra with multiplication rules (a factor set) given by the c(σ, τ). So A is a simple algebra that represents the image of \overline{c} in Br(C_{k}). By comparing dimensions one finds A = E. One concludes that E ≅ Matr(m, F^{o}) for some m where F^{o} is the (skew) field associated to \overline{c}. Further C_{K} = End_{K[∂]}(M) consists of all elements of E which commute with C_{K}. Hence C_{K} is a maximal commutative subfield of E. If C_{k} ⊂ ℓ ⊂ C_{K} then part (e) follows from the statement that ℓ ⊃ C_{k} is a splitting field for F^{o} if and only if the image of \overline{c} in H^{2}(Gal(C_{K}/ℓ), C_{K}^{∗}), under the restriction map, is 1. If ℓ is not a subfield of C_{K}, then replace C_{K} by the normal closure of ℓ · C_{K}. Then the same proof applies; assumption (i) still holds by lemma 2.3 (b), assumption (ii) is clear. □

Corollary 2.9 We keep the notations and assumptions of theorem 2.8.

(1) Suppose that M is irreducible as a K-differential module. Then M viewed as k-differential module is the direct sum of m copies of an irreducible differential module N over k with End_{k[∂]}(N) = F^{o}.

(2) Let be given:

∗ a skew field F^{o} of finite dimension over its center C_{k},

∗ m ≥ 1 and a maximal commutative subfield C_{K} of Matr(m, F^{o}), which is Galois over C_{k},

∗ an irreducible k-differential module N with End_{k[∂]}(N) = F^{o}. Then M, the direct sum of m copies of N, has a natural structure of an irreducible K = C_{K}(x)-differential module that satisfies the assumptions of theorem 2.8.

Proof. (1) Let P_{i} ∈ Matr(m, F^{o}) = E = End_{k[∂]}(M) denote the matrix with all entries 0 with the exception of an entry 1 at the position (i, i). Put N_{i} = P_{i}M. Then N_{i} is a k-submodule of M and M = ⊕_{i}N_{i}. An element L ∈ End_{k[∂]}(N_{i}) can be extended to an element L̃ ∈ E by prescribing L̃ = 0 on each N_{j} with j ≠ i. The structure of E implies that L̃ is a diagonal matrix with zeros on the diagonal, except for the position (i, i). Hence L ∈ F^{o}. Since the matrices P_{i} are conjugated in Matr(m, C_{k}), the k-differential modules N_{i} are all isomorphic. Moreover, since N_{i} is semi-simple and End_{k[∂]}(N_{i}) = F^{o} one has that N_{i} is irreducible.

(2) M = N ⊕ ⋯ ⊕ N has End_{k[∂]}(M) = Matr(m, F^{o}). The embedding of C_{K} in Matr(m, F^{o}) makes M into a K-differential module. End_{K[∂]}(M) consists of the elements in Matr(m, F^{o}) commuting with C_{K}. Thus End_{K[∂]}(M) = C_{K}. Further M is a semi-simple K-differential module since it is a semisimple k-differential module. Thus M is an irreducible K-differential module. Finally, by the Skolem-Noether theorem ([Bl], Théorème III-4, or [Rei], (7.21)) for every σ in the Galois group of C_{K}/C_{k}, there is an invertible element Φ(σ) ∈ Matr(m, F^{o}) with σ(λ) = Φ(σ)λΦ(σ)^{−1} for all λ ∈ C_{K}. Then Φ(σ) : M → M is a σ-linear map which commutes with ∂. □

For a general finite Galois extension of differential fields k ⊂ K with group G, the situation is more complicated; e.g. one-dimensional modules need not descend, and lemma 2.3 no longer applies. Parts (a) and (b) of the following proposition can be proved along the lines of the proof of theorem 2.8. Part (c) follows from 2.11 (a) below, and part (d) follows from 2.11 (c).

Theorem 2.10 The associated 2-cocycle in the general case. Let K/k be a Galois extension of differential fields with Galois group G. Let C_{K} denote field of constants of K. Let a differential module M over K of dimension n satisfy:

(i) the group of the differential automorphisms of M is C_{K}^{∗}, and (ii) for every σ ∈ G one has ^{σ}M ≅ M.

(a) These data determine a 2-cocycle class \overline{c} ∈ H^{2}(G, C_{K}^{∗}) of an order d, dividing [K : k] if K is finite over k. Moreover M descends to k if and only if \overline{c} = 1.

(b) The tensor product M ⊗ ⋯ ⊗ M of d copies of M has associated 2cocycle \overline{c}^{d} and therefore this tensor product descends to k. The same holds for sym^{d}M, the symmetric tensor product of d copies of M.

(c) The image of \overline{c} under the map H^{2}(G, C_{K}^{∗}) → H^{2}(G, K^{∗}) defines a (skew) field F with center k. Let k ⊂ ℓ ⊂ K. Then ℓ is a splitting field for F if and only if there exists a module L of dimension 1 over K such that ^{σ}L ≅ L for all σ ∈ Gal(K/ℓ) and L ⊗ M descends to ℓ.

(d) If K = \overline{k}, and F, ℓ as in part (c), then ℓ is a splitting field if and only if M descends to ℓ.

The condition ^{σ}L ≅ L in (c) is irrelevant when M is not cyclic-imprimitive (for a definition see section 4, see also theorem 4.4).

Consider differential modules M of dimension 1 over K such that ^{σ}M ≅ M for all σ ∈ G. The isomorphism classes of these modules form an abelian group I with the tensor product as multiplication. Let I_{0} denote the subgroup of the classes of differential modules which descend to k.

Proposition 2.11 Let C_{K} denote the constant field of K. Then

(a) I/I_{0} is naturally isomorphic to the kernel of H^{2}(G, C_{K}^{∗}) → H^{2}(G, K^{∗}).

(b) Let M = Ke represent an element of I. Then E := End_{k[∂]}(M) is a semi-simple algebra over C_{k}. If K = C_{K} · k, then E is a central simple algebra over C_{k}. Moreover, M descends to ℓ with k ⊂ ℓ ⊂ K if and only if the field of constants C_{ℓ} is a splitting field for E. If K ≠ C_{K} · k, then, in general, the 2-cocycle in H^{2}(G, C_{K}^{∗}) attached to M, does not describe a central simple algebra over C_{k} or k. Moreover, E is (in general) not a central simple algebra over C_{k}.

(c) If K = \overline{k} then I = I_{0}.

Proof. (a) For any differential field L one writes Q(L) for the isomorphism classes of the 1-dimensional differential modules over L. The tensor product makes Q(L) into a commutative group and there is an exact sequence

0 → L^{∗}/C_{L}^{∗} → L → Q(L) → 0,

where C_{L} denotes the field of constants of L; the first non trivial arrow is defined by f ↦ \frac{f′}{f} and the second non trivial arrow maps f ∈ L to the isomorphism class of the differential module (Le, ∂) with ∂e = fe. We consider this exact sequence with L = K and the exact sequence 0 → C_{K}^{∗} → K^{∗} → K^{∗}/C_{K}^{∗} → 0. These sequences of G-modules induce the usual long exact sequences. Using that H^{1}(G, K) = 0 and H^{1}(G, K^{∗}) = 0 one obtains exact sequences

0 → H^{1}(G, C_{K}^{∗}) → Q(k) → Q(K)^{G} → H^{1}(G, K^{∗}/C_{K}^{∗}) → 0,
0 → H^{1}(G, K^{∗}/C_{K}^{∗}) → H^{2}(G, C_{K}^{∗}) → H^{2}(G, K^{∗}).

Clearly I = Q(K)^{G} and I_{0} is the image of Q(k) → Q(K)^{G}.

(c) If K = \overline{k}, then the group K^{∗}/C_{K}^{∗} is a vector space over Q and hence H^{1}(G, K^{∗}/C_{K}^{∗}) = 0.

(b) The algebra {∑_{σ∈G} m_{σ}Φ(σ)| m_{σ} ∈ K} is equal to the algebra of all k-linear endomorphisms of M. This follows from the K-linear independence of the maps {σ : K → K| σ ∈ G}. One derives from this that E = {∑_{σ∈G} c_{σ}Φ(σ)| c_{σ} ∈ C_{K}}. If K = C_{K} · k, then G is the Galois group of C_{K}/C_{k}. Again, by [Bl], Proposition IV-1, one has that E is a central simple algebra over C_{k} of dimension [C_{K} : C_{k}]^{2}. Consider an intermediate field ℓ. Then K = C_{K} · ℓ. So K/ℓ and C_{K}/C_{ℓ} have the same Galois group H. Now M descends to ℓ if and only if the restriction of the 2-cocycle \overline{c} to H is trivial, if and only if C_{ℓ} is a splitting field for E. Suppose that K ≠ C_{K} ·k. Then C_{K}/C_{k} is still a Galois extension but its Galois group is different from G. The 2-cocycle in H^{2}(G, C_{K}^{∗}), attached to M, need not describe a central simple algebra. Consider the example (4) of 2.5. The algebra E for this example is isomorphic to Q(i). It is also interesting to make part (a) explicit for this example. One has that I/I_{0} is isomorphic to the kernel of H^{2}({1, σ}, Q^{∗}) → H^{2}({1, σ}, Q(t)^{∗}). One can verify that this kernel is the group of two elements. The example describes the non-trivial element in this kernel. □

Remarks: Assume K = C_{K} · k. Then theorem 2.8 is valid for a differential field k precisely when I = I_{0}. The proof of 2.8 part (b) shows I = I_{0} for k = C_{k}(x). A similar argument shows that if k is the function field of a nonsingular algebraic curve with a point defined over C_{k} then I = I_{0}. In subsection 2.5 we will show that proposition 2.11 is related to Amitsur’s construction.

2.4 Differential modules over a skew differential field

In the sequel of this section we will produce explicit examples for corollary 2.9. More precisely, for a given finite Galois extension C_{k} ⊂ C_{K} and a skew field F^{o} of finite dimension over its center C_{k} such that C_{K} is a maximal commutative subfield of Matr(m, F^{o}), we will produce an irreducible differential module M over K = C_{K}(x) which satisfies (i) and (ii) of 2.8 and such that End_{k[∂]}(M) = Matr(m, F^{o}). The basic feature of the construction is the introduction of differential modules over skew differential fields. By 2.9, M corresponds to a k[∂]-module N with End_{k[∂]}(N) = F^{0}. So N is a k[∂]-module as well as an F^{0}-module. Then N is an F [∂]-module where F is defined in 2.13 below. Hence, every example for 2.8/2.9 comes from a differential module over a skew field.

Definitions 2.12 Let k be a differential field and F a skew field of finite dimension over its center k. A differentiation f ↦ f′ on F is an additive map from F to itself such that (fg)′ = f′g + fg′ for all f, g ∈ F and such that the restriction of f ↦ f′ to k ⊂ F is the differentiation of k. A differential module M over F is a finite dimensional left vector space over F, equipped with an additive map ∂ : M → M satisfying ∂(fm) = f′m + f∂m for all f ∈ F and m ∈ M.

Let e_{1}, …, e_{n} be a basis of M over F. Then ∂ is determined by the elements ∂e_{1}, …, ∂e_{n}. Moreover, these elements in M can be chosen arbitrarily.

The next proposition together with part (2) of corollary 2.9 provides the required examples.

Proposition 2.13 Let F^{o} be a skew field of finite dimension over its center C_{k}. On the skew field F = F^{o}(x) := F^{o} ⊗_{C_{k}} C_{k}(x) with center k = C_{k}(x), a differentiation is given by (f ⊗ a)′ = f ⊗ a′ for all f ∈ F^{o} and a ∈ k. Let N be a finite dimensional left vector space over F. Then N can be given the structure of differential module over F such that N is irreducible as a k-differential module and End_{k[∂]}(N) = F^{o}.

Proof. Choose a maximal commutative subfield C of F^{o} containing C_{k}. Then N is also a vector space over C(x). Write C = C_{k}(α) and let P ∈ C_{k}[x] denote the monic minimal polynomial of α. The completion of the local ring C_{k}[x]_{(P)} is denoted by O_{L}. The residue field of O_{L} is C. There is a unique subfield of O_{L} containing C_{k}, which maps bijectively to the residue field C. We will identify C with this subfield of O_{L}. After this identification, the element t = x − α ∈ O_{L} is a generator of the maximal ideal. Thus we can identify O_{L} with C[[t]]. The embedding C_{k}[x] → O_{L} has dense image. Let L denote the field of fractions of O_{L}. Then L is the completion of the field k = C_{k}(x) w.r.t. the valuation associated to the irreducible polynomial P.

Let V = End_{F} (N) and W = End_{C(x)}(N). The natural k-algebra homomorphism ϕ : C(x) ⊗_{k} V → W is bijective. Indeed, the first algebra is simple and the two algebras have the same dimension over k. Consider a property (*) of elements B ∈ W which is preserved for all B̃ such that B̃ is close to B w.r.t. a metric on W induced by the embedding W ⊂ End_{L}(L ⊗_{C(x)} N). Then, since k is dense in C(x), there exists an element in V with property (*).

We identify N with F^{a} and define the standard differentiation on N by n = (n_{1}, …, n_{a}) ↦ n′ = (n′_{1}, …, n′_{a}). A structure ∂ of N as a differential module over F has the form ∂n = n′ + A(n) with A ∈ V. A structure ∂ on N as a differential module over C(x) = C(t) has the form ∂n = n′ + B(n) with B ∈ W. Property (*) is defined by: the Newton polygon of a cyclic element of the module C((t)) ⊗_{C(t)} N has slope \frac{1}{r}, where r is the dimension of N as C(x)-vector space. This property implies that N̂ := \overline{C}((t)) ⊗_{C(t)} N is irreducible and End_{\overline{C}((t))[∂]}(N̂) = \overline{C}. Thus N is an irreducible differential module over C(x) and End_{C(x)}(N) = C. Clearly (*) is preserved under a small perturbation w.r.t. a metric on W induced by the embedding W ⊂ End_{L}(L ⊗_{C(x)} N). We conclude that: N can be given the structure of a differential module over F such that N and \overline{C}(x) ⊗_{C(x)} N are irreducible differential modules over C(x) and \overline{C}(x). Moreover End_{\overline{C}(x)[∂]}(\overline{C}(x) ⊗_{C(x)} N) = \overline{C}.

Let C̃ ⊃ C be the normal closure of C ⊃ C_{k}. Put Ñ = C̃(x) ⊗_{C(x)} N. As a k-differential module, Ñ is isomorphic to a direct sum of [C̃ : C] copies of N. Therefore, End_{k[∂]}(Ñ) is equal to Matr([C̃ : C], End_{k[∂]}(N)) and contains the simple algebra A := Matr([C̃ : C], F^{o}) with center C_{k}. Further C̃ is a maximal commutative subfield of A (since it has the correct dimension). By the Skolem-Noether theorem, every σ ∈ Gal(C̃/C_{k}) extends to an inner automorphism of A. Thus the C̃(x)-module Ñ satisfies conditions (ii) of 2.8. Further Ñ is irreducible and End_{C̃(x)[∂]}(Ñ) = C̃. Condition (i) of 2.8 holds and we can apply the first part of 2.9. One has End_{k[∂]}(Ñ) = Matr(b, F^{o}) for some b ≥ 1. Now C̃ is also a maximal commutative subfield of End_{k[∂]}(Ñ). Therefore the latter algebra coincides with A and End_{k[∂]}(N) = F^{o}. Finally, since N is semi-simple as a differential module over k, it follows that N is irreducible. □

We now give a second construction of examples for corollary 2.9 with m = 1, which will produce nicer examples because there will be only one irregular singularity (at x = ∞). Given is a skew field F^{o} of finite dimension over its center C_{k} and a maximal commutative subfield C_{K} of F^{o} which is a Galois extension of C_{k}. As before F = F^{o}(x) = F^{o} ⊗_{C_{k}} C_{k}(x) is made into a skew differential field by the formula (f ⊗ a)′ = f ⊗ a′ for all f ∈ F^{o} and a ∈ C_{k}(x). Then K = C_{K}(x) is a maximal commutative subfield of F and the restriction of the differentiation of F to K is the obvious differentiation of K.

Proposition 2.14 We use the above notations. Let M be a finite dimensional left vector space over F. Then M can be given a structure of differential module over F such that M is an irreducible K-differential module with the properties:

(a) End_{K[∂]}(M) = C_{K}.

(b) ^{σ}M ≅ M for all σ ∈ Gal(K/k).

(c) End_{k[∂]}(M) = F^{o} and M is irreducible as k-differential module.

(d) \underline{M} := \overline{C_{K}}(x) ⊗_{K} M is an irreducible differential module over \overline{C}_{K}(x).

(e) End_{\overline{C}_{K}(x)[∂]}(\underline{M}) = \overline{C}_{K}.

Proof. As in the proof of proposition 2.13, we identify M with F^{a} for some a ≥ 1 and define the operation ′ on F^{a} by (f_{1}, …, f_{a})′ = (f′_{1}, …, f′_{a}). A structure ∂ of differential module over F on M is given by ∂m = m′ + mD, with D ∈ Matr(a, F).

For D we make the choice D = ∑_{i=0}^{N} D_{i}x^{i}, with:

(1) D_{N} is a diagonal matrix with distinct coefficients in C_{k} on the diagonal,

(2) {D_{0}, …, D_{N−1}} are generators of Matr(a, F^{o}) over C_{k},

(3) M is irreducible as a differential module over F. For a = 1, property (3) is obvious. Moreover in that case we may replace (1)+(2) by: D_{0}, …, D_{N} generate F^{o} over C_{k}. If a > 1, then one could apply the method of proposition 2.13 in order to obtain (3) and (a–e), but we will avoid this because it leads to examples more complicated than ∑_{i=0}^{N} D_{i}x^{i}, and because (3) is easy to verify for our explicit examples.

(i) End_{k[∂]}(M) = F^{o}. For any algebra B we write B^{opp} for the opposite algebra. For any algebra B we write Matr(a, B) for the algebra of the a × a-matrices with coordinates in B. Now F and Matr(a, F)^{opp} are central simple algebras over k. By [Ren], Corollaire 4, p 107, the algebra F ⊗_{k} Matr(a, F)^{opp} is again simple. This algebra is mapped to End_{k}(F^{a}), the algebra of the k-linear endomorphisms of F^{a}, by the following formula (f ⊗ B)(v) = fvB for f ∈ F, v ∈ F^{a}, B ∈ Matr(a, F)^{opp}. Since the first algebra has only trivial two-sided ideals, this map is injective. By counting dimensions over k, one concludes that the map is bijective. Let b_{1}, …, b_{d^{2}} denote a basis of F^{o} over C_{k}. From the above it follows that every k-linear map L : M → M can uniquely be written as L(v) = ∑_{i=1}^{d^{2}} b_{i}vA_{i}, with all A_{i} ∈ Matr(a, F).

For L as above one calculates that (∂L − L∂)v = ∑_{i=1}^{d^{2}} b_{i}vB_{i}, where B_{i} equals the matrix A′_{i} + A_{i}D − DA_{i}. Hence L commutes with ∂ if and only if A′_{i} + A_{i}D − DA_{i} = 0 for every i.

Suppose that A ∈ Matr(a, F) is a non-zero solution of A′ = [D, A]. Let q ∈ C_{k}[x] denote the monic polynomial of minimal degree such that qA ∈ Matr(a, F^{o}[x]). Write A = q^{−1}B, then −q′q^{−2}B +q^{−1}B′+q^{−1}BD−q^{−1}DB = 0. After multiplying this identity with q one finds that q′A ∈ Matr(a, F^{o}[x]). One concludes that q = 1. Hence A ∈ Matr(a, F^{o}[x]).

The set V of all solutions A ∈ Matr(a, F^{o}[x]) of A′ = [D, A] is an algebra over C_{k}. Moreover, V is a finite dimensional vector space over C_{k} since A′ = [D, A] can be interpreted as a linear differential equation over the differential field k. Any A ∈ V, A ≠ 0 can be written in the form A = ∑_{i=0}^{e} A_{i}x^{i} with all A_{i} ∈ Matr(a, F^{o}) and A_{e} ≠ 0. One finds that [D_{N}, A_{e}] = 0 and thus A_{e} is a diagonal matrix (with coefficients in F^{o}). The elements 1, A, A^{2}, A^{3}, … all belong to V. In particular, there is a non-trivial relation λ_{0}1 + λ_{1}A + ⋯ + λ_{s}A^{s} = 0 with all λ_{i} ∈ C_{k} and λ_{s} ≠ 0. If e > 0, then, since A_{e} is a non-zero diagonal matrix, the term x^{es} is present in A^{s} and not in the A^{i} with i < s. This contradiction implies that e = 0. Further A = A_{0} lies in the center of Matr(a, F^{o}), since A_{0} commutes with all D_{i}. Hence A = λ1 for some λ ∈ C_{k}. Hence End_{k[∂]}(M) = F^{o}.

(ii) End_{K[∂]}(M) = C_{K} and M is an irreducible differential module over K. The first statement holds because C_{K} is a maximal commutative subfield of F^{o}. Let 0 ≠ N ⊂ M be an irreducible K-differential module. For any f ∈ F^{o}, f ≠ 0, also fN is an irreducible K-differential module. The sum ∑ fN, where f runs in a basis of F^{o} over C_{K}, is a semi-simple K-differential module. Since this object is invariant under left multiplication by F, and M is an irreducible F -differential module, one has ∑ fN = M. So M is semisimple over K and since End_{K[∂]}(M) contains only the trivial idempotents it follows that M is irreducible over K. The same argument (or 2.7) shows that M is semi-simple and irreducible over k.

(iii) ^{σ}M ≅ M as differential modules over K. The Skolem-Noether theorem asserts that for σ ∈ G there exists non-zero f_{σ} ∈ F^{o} such that σ(λ) = f_{σ}λf_{σ}^{−1} for every λ ∈ C_{K}. One defines Φ(σ)(m) = f_{σ}m. Clearly Φ(σ) : M → M is a σ-linear bijection, commuting with ∂. This proves (b). Statements (d) and (e) follow from 2.7 and 2.3. □

Examples 2.15 Skew fields over Q.

(1) Let H = Q + Qi + Qj + Qk denote Hamilton’s quaternion field over Q. We consider a maximal commutative subfield C_{K} = Q(i) and the fields K := C_{K}(x), k = Q(x). One provides the 1-dimensional left vector space M = H(x)e over H(x) with ∂ defined by ∂e = de for some d ∈ H(x). According to proposition 2.14, the choice d = i+jx makes M into an example for 2.8 and 2.9. Let σ be the non-trivial element in Gal(K/k). Then Φ(σ), defined by Φ(σ)he = jhe for all h ∈ H(x), is a good choice for the σ-linear bijection commuting with ∂. We note that the 2-cocycle c has the form c(1, 1) = c(1, σ) = c(σ, 1) = 1 and c(σ, σ) = −1.

(i) Explicit formulas. e ∈ M is cyclic for M as a differential module over k = Q(x). The minimal monic operator L_{4} ∈ k[∂] with L_{4}e = 0 has the form

L_{4} = ∂^{4} + (2 + 2x^{2})∂^{2} + 4x∂ + (4 + 2x^{2} + x^{4}).

By 2.14 (c), L_{4} is irreducible as an element of Q(x)[∂]. Moreover, e ∈ M is also cyclic for M as Q(i)(x)[∂]-module. The minimal monic operator L_{2} ∈ Q(i)(x)[∂] has the form

L_{2} = ∂^{2} − x^{−1}∂ + (ix^{−1} + 1 + x^{2}).

L_{2} must be a right hand factor of L_{4} in Q(i)(x)[∂]. Indeed,

L_{4} = (∂^{2} + x^{−1}∂ + (−x^{−2} − ix^{−1} + 1 + x^{2})) · L_{2}.

By 2.14 (d), the operator L_{2} is irreducible as an element of \overline{Q}(x)[∂]. As we know L_{2} is equivalent to its conjugate ∂^{2} − x^{−1}∂ + (−ix^{−1} + 1 + x^{2}) and L_{2} is not equivalent to any second order differential operator in Q(x)[∂]. Now Q(\sqrt{−m}) is a splitting field of H when m > 0 is the sum of three squares in Q. Hence for every such m, L_{2} must be equivalent to an operator in Q(\sqrt{−m})(x)[∂], and L_{4} can be factored as a product of two irreducible operators in Q(\sqrt{−m})(x)[∂].

(ii) The associated third order operator L_{3} ∈ Q(x)[∂]. According to 2.10 (b), the symmetric square N = sym^{2}M of M as Q(i)(x)[∂]- module descends to Q(x). We want to make this explicit for our example. The σ-linear map Ψ(σ) : sym^{2}M → sym^{2}M is defined by

Ψ(σ)(m_{1} ⊗ m_{2}) = Φ(σ)m_{1} ⊗ Φ(σ)m_{2} = jm_{1} ⊗ jm_{2}.

In particular, Ψ(σ)^{2} is the identity. Let N^{0} denote the set of the elements of N invariant under Ψ(σ). Give M the basis e, je. Then N has Q(i)(x)-basis e ⊗ e, je ⊗ e, je ⊗ je. One finds that N^{0} has Q(x)-basis e ⊗ e + je ⊗ je, ie ⊗ e − ije ⊗ je, ije ⊗ e. Further N^{0} is a differential module over Q(x), since Ψ(σ) commutes with ∂. Thus N = Q(i)(x) ⊗_{Q(x)} N^{0}. We take ije ⊗ e as cyclic element of N^{0} and let L_{3} be its minimal operator: L_{3}(ije ⊗ e) = 0. A calculation shows that

L_{3} = ∂^{3} − 2x^{−1}∂^{2} + (2x^{−2} + 4 + 4x^{2})∂ + 4x.

L_{3} must be equivalent to the symmetric square of the operator L_{2} above. The latter does not have coefficients in Q(x) because it corresponds to the cyclic vector e ⊗ e, which is not invariant under Ψ(σ).

Since N is a symmetric square, we see that N^{0} becomes a symmetric square after making a suitable algebraic extension C of Q. The fields C are precisely the splitting fields for H. We will return to this in the next section.

(2) Example for a general quaternion field over Q. If we repeat the computation with i replaced by b_{1}, j replaced by b_{2}, with the following rules of multiplication: b_{1}^{2} = A_{1}, b_{2}^{2} = A_{2}, b_{2}b_{1} = −b_{1}b_{2}, replacing H by F^{o} = Q + Qb_{1} + Qb_{2} + Qb_{1}b_{2}, and d by b_{1} + b_{2}x then we find the following L_{2}

L_{2} = ∂^{2} − x^{−1}∂ + \sqrt{A_{1}}x^{−1} − A_{1} − A_{2}x^{2}.

Now the splitting fields of F^{o} are precisely the fields C for which the conic

A_{1}X^{2} + A_{2}Y^{2} − 1 = 0

has a point (X, Y) ∈ C^{2}. In particular, F^{o} is a skew field if and only if this conic has no Q-rational point. For example if A_{1} = 2 and A_{2} = 3, then F^{o} is a skew field with center Q. We note that the following operator

L′_{2} = ∂^{2} − A_{2}x^{2} − A_{1} + \sqrt{A_{2}}

is equivalent to L_{2}. Hence it defines the same descent problem, and L′_{2} descends to C(x)[∂] if and only if C is a splitting field of F^{o}. The symmetric square of L_{2} is equivalent to

L_{3} = ∂^{3} − 2x^{−1}∂^{2} + (2x^{−2} − 4A_{1} − 4A_{2}x^{2})∂ − 4A_{2}x

which is equivalent to

∂^{3} − 4(A_{1} + A_{2}x^{2})∂ − 12A_{2}x.

Suppose A_{1}, A_{2} are integers, and that the conic has a rational point (X, Y) ∈ Q^{2}. Then L_{2} descends to Q(x) and the cocycle class \overline{c} in theorem 2.8 is 1. If both parts (i) and (ii) of the descent data are explicitly known, then one can explicitly calculate descent: The module N in 2.2 part (3) can be found as {∑_{σ∈G} Φ(σ)(m)|m ∈ M}. Conversely, if one knows an explicit descent, then descent data can also be explicitly calculated. Now suppose that A_{1}, A_{2} are given, but X, Y are not. It is easy to calculate part (i) of the descent data for the example L_{2}. However, to calculate part (ii) of the descent data, one must multiply ϕ(σ) by a suitable element of C_{K}. This means solving a norm equation, which is equivalent to finding a rational point on the conic. Known algorithms for finding rational points on a conic use factorization of integers. So finding a rational point can be computationally hard (but only if A_{1}, A_{2}, −A_{1}A_{2} are not squares, and at least one A_{i} is hard to factor in Z, for an example see http://www.math.fsu.edu/~hoeij/files/conic). For such A_{1}, A_{2}, finding part (ii) of the descent data for L_{2}, or equivalently, finding descent, is computationally hard. Finding an irreducible submodule of M, viewed as a differential module over Q(x), is then also computationally hard, so factoring

L_{4} = ∂^{4} − 2(A_{2}x^{2} + A_{1})∂^{2} − 4A_{2}x∂ + A_{1}^{2} − 3A_{2} + A_{2}^{2}x^{4} + 2A_{1}A_{2}x^{2}

in Q(x)[∂] is hard. Indeed, one can parametrize all monic second order factors

∂^{2} − s/(t + sx) ∂ − A_{2}x^{2} − A_{1} − u/(t + sx)

of L_{4} in terms of points (s : t : u) on the conic A_{1}s^{2} + A_{2}t^{2} − u^{2} = 0, and hence finding a second order factor is equally hard as finding a point on the conic.

(3) Example with M of dimension 2 over F. Consider the quaternion field F^{o} = Q + Qb_{1} + Qb_{2} + Qb_{1}b_{2}, with b_{1}^{2} = 2, b_{2}^{2} = 3, b_{2}b_{1} = −b_{1}b_{2}. Let F = F^{o}(x) and M = F^{2} be the 2-dimensional differential module over F given by the following action of ∂ (recall that we are using row notation v = (v_{1}, v_{2}))

∂v = v′ + v ( \begin{matrix}0 & 1 \\ b_{1} + b_{2}x & 0 \end{matrix} ) .

End_{Q(x)[∂]}(M) = F^{o}, see 2.14. M is irreducible as Q(\sqrt{2})(x)-differential module. The cyclic vector (1, 0) gives the following operator:

∂^{4} − 2x^{−1}∂^{3} + 2x^{−2}∂^{2} + 2\sqrt{2}x^{−1}∂ − 2\sqrt{2}x^{−2} − 2 − 3x^{2}.

This operator is irreducible (even over \overline{Q}) and descends to C(x)[∂] if and only if C is a splitting field for F^{o}.

(4) A skew field F^{o} of dimension 9 over Q. Let α be a solution of α^{3} − 3α − 1 = 0. Then Q(α) is Galois over Q with Galois group generated by σ, where σ maps α to 2 − α^{2}. Now take b such that bα = σ(α)b and b^{3} = 2. Let F^{o} be the skew field generated by α and b, take d = b + αx, let M = F^{o}(x), and ∂v = v′ + vd. Then we find the following operator

∂^{3} + (α^{2} − 2α − 2 − 3x^{2})∂ − x^{3} − (1 + α + α^{2})x − 2 ∈ Q(α)[∂].

It is irreducible, even as an element of \overline{Q}(x)[∂], and descends to C(x)[∂] if and only if C is a splitting field for F^{0}.

2.5 Amitsur’s construction

Let k be a differential field of characteristic 0, having C_{k} as field of constants. Amitsur considers an irreducible differential module M of dimension n over k with the property that M^{∗} ⊗ M is a trivial differential module. In other words, Hom(M, M) is a trivial differential module and the ring of endomorphisms E := End_{k[∂]}(M), which equals ker(∂, Hom(M, M)), has dimension n^{2} over C_{k}. Now k ⊗_{C_{k}} E is the k-algebra of all k-linear maps M → M. It follows that E is a skew field with center C_{k} and that k is a splitting field for E. If C_{K} ⊃ C_{k} is a finite extension and a splitting field for E, and K = C_{K} · k, then the differential module K ⊗_{k} M is a direct sum of copies of a 1dimensional differential module over K. Indeed, End_{K[∂]}(K ⊗_{k} M) is isomorphic to the matrix algebra Matr(n, C_{K}).

One of the main results, Theorem 16 of [A] is: Any skew field E of dimension n^{2} over its center C_{k} that has k as splitting field is obtained in this way.

We sketch the proof of this result. Put M = k^{n}. By definition there is a homomorphism a ∈ E ↦ P_{a} ∈ End_{k}(M). One defines ∂_{0} : M → M by ∂_{0}(f_{1}, ⋯, f_{n}) = (f′_{1}, ⋯, f′_{n}). For any k-linear map L : M → M one defines the k-linear map L′ by L′ = ∂_{0}L − L∂_{0}. The two representations

E → End_{k}(M ⊕ M), given by a ↦ ( \begin{matrix}P_{a} & 0 \\ 0 & P_{a} \end{matrix} ) and a ↦ ( \begin{matrix}P_{a} & P′_{a} \\ 0 & P_{a} \end{matrix} ), are

isomorphic. Indeed, every finitely generated left module over a semi-simple algebra is itself semi-simple. Using this one obtains a Q ∈ End_{k}(M) with the property P′_{a} = QP_{a} − P_{a}Q for all a ∈ E. Define now ∂ : M → M by ∂ = ∂_{0} − Q. This makes M = (M, ∂) into a differential module over k and there is a homomorphism E → End_{k[∂]}(M). Hence Hom(M, M) is a trivial differential module and End_{k[∂]}(M) = E. This is Amitsur’s construction.

The differential field k = C_{k}(x) does not produce an example for Amitsur’s theorem, since it is not a splitting field for any non-trivial skew field of finite dimension over its center C_{k}. Consider the differential field k = Q(s, t) with s^{2} + t^{2} = −1 and s′ = 1, t′ = −st^{−1}. Then k is a splitting field for Hamilton’s quaternions H = Q + Qi + Qj + Qk over C_{k} = Q. One

defines a ∈ H ↦ P_{a} ∈ End_{k}(M) with M = k^{2} by P_{i} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and

P_{j} = ( \begin{matrix}s & t \\ t & −s \end{matrix} ). One calculates that Q = ( \begin{matrix}0 & 1/2t \\ −1/2t & 0 \end{matrix} ) and that the

minimal monic operator that annihilates the cyclic vector ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ M is

L = (s^{2} + 1)∂^{2} + s∂ − \frac{1}{4}. Let m be a positive squarefree integer and K = k(\sqrt{−m}). From the above it follows that L is irreducible as an element of k[∂], and that L factors in K[∂] if and only if m is the sum of ≤ 3 squares in Q.

Note that Amitsur’s theorem follows from proposition 2.11 (a). Consider a differential field k with field of constants C_{k} and a skew field E of dimension n^{2} over its center C_{k}, such that k is a splitting field for E.

Let C_{K} be a splitting field for E, and a finite Galois extension of C_{k}. Let G be the Galois group of C_{K}/C_{k}. Then [E] ∈ Br(C_{K}/C_{k}) ≅ H^{2}(G, C_{K}^{∗}). Now G is also the Galois group of K/k, where K = C_{K} · k, so H^{2}(G, K^{∗}) ≅ Br(K/k). Since k is a splitting field, [E ⊗ k] is trivial, hence [E] is in the kernel of H^{2}(G, C_{K}^{∗}) → H^{2}(G, K^{∗}). According to proposition 2.11 (a), [E] is the image of some N ∈ I = Q(K)^{G}. This N is a 1-dimensional differential module over K such that ^{σ}N ≅ N for all σ ∈ G. Let M be N considered as a differential module over k. Explicit construction of the map Q(K)^{G} → H^{2}(G, C_{K}^{∗}) shows that [End_{k[∂]}(M)] = [E]. Then M has a submodule M′ of dimension n over k (if C_{K} is a maximal commutative subfield of E then M = M′), and End_{k[∂]}(M′) = E. Since N is unique up to I_{0}, see proposition 2.11 (a), the isomorphism class of M′ is unique up to tensoring with 1-dimensional modules over k. Thus it must correspond to Amitsur’s construction up to this equivalence.

3 Differential modules of dimension 3

In this section we consider a differential operator L_{3} ∈ K[∂] of order 3 over a differential field K. The first question is whether L_{3} is equivalent to the second symmetric power of a differential operator of order 2. Singer (see [Si]) showed that the question has a positive answer if and only if one can produce a certain conic and a K-rational point on this conic. This raises a second question: Which conics can occur? We will use skew differential fields to answer this.

The first question translates in terms of differential modules as follows. A differential module B of dimension 3 over K is given. Is B isomorphic to the second symmetric power sym_{K}^{2}A of a differential module A of dimension 2 over K? We recall that sym_{K}^{2}A is defined as the K-vector space A ⊗^{s}_{K} A with ∂ given by the formula ∂(a_{1} ⊗a_{2}) = (∂a_{1})⊗a_{2} +a_{1} ⊗(∂a_{2}). Our interest in this question lies in the fact that a differential module B may not be a second symmetric power, but could become a second symmetric power after enlarging the field of constants of K. Example 2.15, (1), (ii) has this feature. There is again a 2-cocycle responsible for this phenomenon (see section 4) and we will construct examples using quaternion fields. First we investigate some properties of the second symmetric power.

3.1 Properties of the second symmetric power

Proposition 3.1 Let B be a differential module of dimension 3 over a differential field K. The following are equivalent:

(1) B ≅ sym_{K}^{2}A for some differential module A of dimension 2 over K.

(2) sym_{K}^{2}B has a 1-dimensional submodule L with the following property. Let b_{1}, b_{2}, b_{3} be any basis of B over K. Then L is generated by an element of the form ∑_{1≤i≤j≤3} c_{i,j}b_{i} ⊗b_{j} with c_{i,j} ∈ K and the quadric ∑_{1≤i≤j≤3} c_{i,j}X_{i}X_{j} = 0 in P_{K}^{2} is non-degenerate and has a K-rational point.

Proof. (1)⇒(2). Let a_{1}, a_{2} be a basis of A over K. Put b_{1} = a_{1} ⊗ a_{1}, b_{2} = a_{2} ⊗ a_{2}, b_{3} = a_{1} ⊗ a_{2}. Then the 1-dimensional subspace L of sym_{K}^{2}B with generator b_{1} ⊗ b_{2} − b_{3} ⊗ b_{3} is easily seen to be a differential submodule. Moreover X_{1}X_{2} − X_{3}^{2} = 0 has a non-trivial solution in K^{3}. The reasoning above is based on the observation that the canonical morphism of differential modules ϕ : sym_{K}^{2}B → sym_{K}^{4}A is surjective. Comparing the dimensions, one finds that the kernel of ϕ is a 1-dimensional submodule of sym_{K}^{2}B. (2)⇒(1). The assumptions on the quadric ∑_{1≤i≤j≤3} c_{i,j}X_{i}X_{j} = 0 imply that there exists a linear change of the variables X_{1}, X_{2}, X_{3} which transforms the quadratic form into a multiple of X_{1}X_{2} − X_{3}^{2}. Thus B has a basis b_{1}, b_{2}, b_{3} such that the 1-dimensional K-vector space generated by b_{1} ⊗ b_{2} − b_{3} ⊗ b_{3} of sym_{K}^{2}B is a differential submodule. One considers a 2-dimensional vector space A over K with basis a_{1}, a_{2} and one defines the K-linear bijection ϕ : sym_{K}^{2}A → B by ϕ(a_{1} ⊗ a_{1}) = b_{1}, ϕ(a_{2} ⊗ a_{2}) = b_{2} and ϕ(a_{1} ⊗ a_{2}) = b_{3}. Let the ∂ on B be given by the formula ∂b_{i} = ∑_{j} e_{j,i}b_{j} for i = 1, 2, 3. The Kvector space A is made into a differential module by putting ∂a_{i} = ∑_{j} d_{j,i}a_{j} with d_{1,1} = e_{1,1}/2, d_{2,1} = e_{3,1}/2, d_{1,2} = e_{3,2}/2, d_{2,2} = e_{2,2}/2. Using that ∂(b_{1} ⊗ b_{2} − b_{3} ⊗ b_{3}) is equal to f(b_{1} ⊗ b_{2} − b_{3} ⊗ b_{3}) for some f ∈ K, one can verify that ϕ is an isomorphism of differential modules. □

Some observations: Let C be an algebraically closed field of characteristic 0. Let K = C(x) be the differential field with differentiation f ↦ \frac{df}{dx}. For this field K, one can omit in part (2) of proposition 3.1 the assumption that the quadric has a K-rational point. Indeed, K is a C_{1}-field. The Tannakian equivalence between differential modules over K and finite dimensional Clinear representations of the universal differential Galois group of K leads to the following translation of 3.1 in terms of representations of linear algebraic groups over C:

Let G be a linear algebraic subgroup of GL(W), where W is a vector space of dimension 3 over C. Suppose that sym^{2}W contains a G-invariant line that defines a non-degenerate quadric. Then there exists a linear algebraic group H ⊂ GL(V), where V has dimension 2 over C, such that H/(H ∩{±1}) ≅ G and the two G-modules sym^{2}V and W are isomorphic. A similar result seems to be valid for Galois representations.

Lemma 3.2 Let B be a differential module of dimension 3 over K. If sym_{K}^{2}B has a 1-dimensional submodule such that the corresponding quadratic form is degenerate, then B is reducible.

Proof. Suppose that the quadratic form associated to the 1-dimensional submodule L is degenerate. This degenerate quadratic form has rank 1 or 2. There exists c_{1}, c_{2} ∈ C_{K} and a basis b_{1}, b_{2}, b_{3} of B such that L is generated by b_{1} ⊗ b_{1} (for rank 1) or by c_{1}b_{1} ⊗ b_{1} + c_{2}b_{2} ⊗ b_{2} (for rank 2). Then Kb_{1} or Kb_{1} + Kb_{2} is a differential submodule of B. □

Lemma 3.3 Let A be an irreducible differential module of dimension 2. Put B = sym_{K}^{2}A. Then B is reducible if and only if there is a field extension K̃ ⊃ K of degree two such that K̃ ⊗_{K} A is reducible.

Proof. Because of 2.7 (4), if B is reducible then B has a one-dimensional submodule L. One can choose a basis a_{1}, a_{2} of A such that a generator of L has one of the following forms: a_{1}⊗a_{1}, a_{1}⊗a_{2} or a_{1}⊗a_{1}−fa_{2}⊗a_{2} where f ∈ K is not a square. The first two cases are excluded since A is irreducible. In the last case one puts K̃ = K(\sqrt{f}) and K̃ ⊗_{K} A = K̃(a_{1} +a_{2}\sqrt{f})⊕K̃(a_{1} −a_{2}\sqrt{f}) and thus K̃ ⊗ A is reducible.

Conversely, let K̃ = K(t) with t^{2} = f ∈ K and write σ for the non-trivial element of Gal(K̃/K). Let K̃e be a submodule of K̃ ⊗ A. If K̃σe = K̃e, then σe = ge for some g ∈ K̃^{∗} and gσ(g) = 1. Then g = \frac{h}{σh} for some h ∈ K̃^{∗} and so σ(he) = he. It follows that he ∈ A and Khe is a submodule of A. This is a contradiction since A is irreducible. Thus K̃ ⊗ A is the direct sum of the submodules K̃e and K̃σe. Then a_{1} = e + σe, a_{2} = te − tσe is a basis of A. Finally a_{1} ⊗ a_{1} − f^{−1}a_{2} ⊗ a_{2} generates a submodule of B. □

Remarks. An irreducible differential module A of dimension 2 will be called imprimitive if there exists a quadratic extension K̃ of K such that K̃ ⊗_{K} A is reducible. Otherwise A will be called primitive. A differential module A can be imprimitive for two different reasons. It is possible that A becomes reducible after a quadratic extension of the field of constants of K. In the second case, A remains irreducible after replacing K by \overline{C_{K}}K. The differential Galois group G is defined for the differential module \overline{C_{K}}K ⊗ A. Imprimitive is now equivalent to: the action of G on the two-dimensional solution space V is irreducible and there are lines L_{1}, L_{2} ⊂ V such that V = L_{1} ⊕ L_{2} and any g ∈ G permutes the lines L_{1}, L_{2}. In other words, G is contained in the infinite dihedral group {g ∈ GL_{2}| {gL_{1}, gL_{2}} = {L_{1}, L_{2}}}.

Lemma 3.4 Suppose that the two-dimensional differential module A is irreducible and primitive. Put B = sym_{K}^{2}A. There are two possibilities:

(1) sym_{K}^{2}B has three distinct submodules of dimension 1. Each one of them determines a non-degenerate quadric having a K-rational point.

(2) sym_{K}^{2}B has a unique submodule of dimension 1.

Proof. After replacing A by L ⊗ A for a suitable 1-dimensional differential module, we may suppose that the differential Galois group G of A is an irreducible, primitive subgroup of SL_{2}(\overline{C_{K}}). We note that the differential Galois group is only well defined over the differential field \underline{K} := \overline{C_{K}}K. Let \underline{A} and \underline{B} denote \underline{K} ⊗ A and \underline{K} ⊗ B. If G is not the group A_{4}^{SL_{2}} (see remarks 4.6 part (3) for a list of groups), then the symmetric power sym4 ; \underline{K} \underline{A} has no 1dimensional submodule and so the same holds for sym_{K}^{4}A. It follows that the only 1-dimensional submodule of sym_{K}^{2}B is the kernel of the canonical morphism sym_{K}^{2}B → sym_{K}^{4}A. Suppose that G is A_{4}^{SL_{2}}. Then by differential Galois theory:

(a) The differential Galois group of \underline{B} is A_{4}.

(b) sym2 ; \underline{K} \underline{B} ≅ M_{0} ⊕ M_{1} ⊕ M_{2} ⊕ \underline{B}, where M_{0}, M_{1}, M_{2} are 1-dimensional

submodules. The differential Galois group of M_{0} is trivial and M_{0} = \underline{L}_{0}, where L_{0} is the kernel of sym_{K}^{2}B → sym_{K}^{4}A.

(c) M_{i} ⊗ \underline{B} ≅ \underline{B} for all i.

(d) The differential Galois group of M_{i}, i = 1, 2 is the quotient C_{3} of A_{4}. Thus sym3 ; \underline{K}M_{i} is a trivial module. Moreover sym2 ; \underline{K}M_{1} ≅ M_{2}. We assume that sym_{K}^{2}B has a 1-dimensional submodule L_{1} ≠ L_{0}. Then \underline{L}_{1} is either M_{1} or M_{2}. We may suppose that \underline{L}_{1} = M_{1}. From the above one concludes that sym2 ; \underline{K}(\underline{L}_{1} ⊗ \underline{A}) is isomorphic to \underline{B} and that the kernel of sym2 ; \underline{K} \underline{B} → sym4 ; \underline{K}(\underline{L}_{1} ⊗\underline{A}) is sym_{K}^{4} \underline{L}_{1} = M_{1}. From 2.3 (a) one concludes that all isomorphisms are defined over the field K. As a consequence M_{2} = \underline{K}⊗L_{2}, where L_{2} := sym_{K}^{2}L_{1}. In the same way, L_{2} is the kernel of the morphism sym_{K}^{2}B ≅ sym_{K}^{2}(L_{2} ⊗ B) → sym^{4}(L_{2} ⊗ A). An application of proposition 3.1 ends the proof. □

3.2 Examples obtained from quaternion fields

Notations and assumptions. F is a quaternion algebra over k = C_{k}(x) with basis b_{0}, …, b_{3}. The multiplication is given by b_{0} = 1, b_{i}^{2} = A_{i} ∈ k = C_{k}(x) for i = 1, 2, 3, b_{1}b_{2} = −b_{2}b_{1} = b_{3}, A_{3} = −A_{1}A_{2}. Then F is a skew field if and only if the equation A_{1}X^{2} + A_{2}Y^{2} − Z^{2} = 0 has only the trivial solution (0, 0, 0) in the field k (see [Bl], Théorème I-5). We will assume that this is the case. Put F_{i} = k(b_{i}) for i = 1, 2, 3. These are maximal commutative subfields of F. As before, a differentiation ′ on F will be a map f ↦ f′ such that (fg)′ = f′g + fg′ for all f, g ∈ F and f′ = \frac{df}{dx} for any f ∈ C_{k}(x) ⊂ F. We note that in general for f ∈ F, the elements f and f′ need not commute, in which case k(f) is not a differential subfield of F. Differentiations on F are not unique, but we can choose one as follows.

Lemma 3.5 There is a unique differentiation ′ on F such that the fields F_{1}, F_{2} are stable under differentiation. The field F_{3} is also stable under this differentiation.

Proof. The condition prescribes b′_{0} = 0 and b′_{i} = \frac{A′_{i}}{2A_{i}} b_{i} for i = 1, 2. From b_{3} = b_{1}b_{2} one deduces that b′_{3} = \frac{A′_{3}}{2A_{3}} b_{3}. Thus (∑_{i=0}^{3} a_{i}b_{i})′ = ∑_{i=0}^{3}(a′_{i} + a_{i} \frac{A′_{i}}{2A_{i}})b_{i}, where A_{0} := 1. On the other hand one can easily verify that this formula defines a differentiation on F. The last statement holds because b′_{3} ∈ F_{3}. □

A 1-dimensional module M = F e over F is described by ∂e = de, where d ∈ F is an arbitrarily chosen element. One can see M as a two-dimensional differential module over the differential field F_{1}. Let σ denote the non-trivial element of the Galois group of F_{1}/k. One considers the σ-linear bijection A(σ) : M → M, defined by A(σ)m = b_{2}m. One has: ∂A(σ) = A(σ)(∂ + \frac{A′_{2}}{2A_{2}} ), or in a shorter notation ∂b_{2} = b_{2}(∂ + \frac{A′_{2}}{2A_{2}} ).

Intermezzo. One has ^{σ}M ≅ L ⊗_{F_{1}} M for the one-dimensional differential module L = F_{1}b over F_{1} with ∂b = \frac{A′_{2}}{2A_{2}} b. Further L ⊗ L is isomorphic to the trivial onedimensional module. As a consequence the 3-dimensional differential module N := sym_{F_{1}}^{2}M descends to k. We will make this explicit by a calculation.

On the differential module N =: sym_{F_{1}}^{2}M of dimension 3 over F_{1} we define an additive operator B(σ) : N → N by the formula B(σ)(m_{1} ⊗ m_{2}) = \frac{1}{A_{2}} b_{2}m_{1} ⊗b_{2}m_{2}. The following properties are easily verified: B(σ) is σ-linear, commutes with ∂, and B(σ)^{2} is the identity. Let N^{0} denote the subset of N consisting of the elements invariant under B(σ). An explicit calculation shows that N^{0} is a vector space of dimension 3 over k with basis n_{1}^{0} = e⊗e+\frac{1}{A_{2}} b_{2}e⊗b_{2}e, n_{2}^{0} = b_{1}e⊗e−\frac{b_{1}}{A_{2}} b_{2}e⊗b_{2}e, n_{3}^{0} = b_{2}e⊗e. Since B(σ) and ∂ commute, N^{0} is a differential module over k and moreover N = F_{1} ⊗_{k} N^{0}. In other words N descends to k.

M, as a differential module over F_{1}, has basis e, b_{2}e. The differential module N has basis n_{1} = e ⊗ e, n_{2} = b_{2}e ⊗ b_{2}e, n_{3} = e ⊗ b_{2}e. By proposition 3.1, sym_{F_{1}}^{2}N has a 1-dimensional submodule, generated by n_{1} ⊗ n_{2} − n_{3} ⊗ n_{3}. This expression can be rewritten in the basis n_{1}^{0}, n_{2}^{0}, n_{3}^{0} and reads

\frac{A_{2}}{4} n_{1}^{0} ⊗ n_{1}^{0} − \frac{A_{2}}{4A_{1}} n_{2}^{0} ⊗ n_{2}^{0} − n_{3}^{0} ⊗ n_{3}^{0}.

Thus sym_{k}^{2}N^{0} contains a one dimensional submodule L and the quadratic form associated to L has the form \frac{A_{2}}{4} X_{1}^{2}−\frac{A_{2}}{4A_{1}} X_{2}^{2}−X_{3}^{2}. This form is equivalent to the quadratic form A_{1}X^{2} + A_{2}Y^{2} − Z^{2} associated to the quaternion field F.

Theorem 3.6 Let F, F_{1}, k, M, N^{0} be as above and let K be an algebraic extension of k.

(1) If K is a splitting field for F then K ⊗_{k} N^{0} is the second symmetric power of some differential module over K.

(2) Assume that K ⊗_{k} N^{0} is the second symmetric power of some differential module over K and that KF_{1} ⊗_{k} N^{0} is irreducible. Then K is a splitting field for F.

Proof. (1) Apply proposition 3.1 to the 1-dimensional submodule K ⊗_{k} L of sym^{2}(K ⊗_{k} N^{0}). By assumption the quadratic form associated to L has a non-zero K-rational point.

(2) By proposition 3.1, sym^{2}(K ⊗_{k} N^{0}) contains a 1-dimensional submodule Z such that the associated quadratic form is non degenerate and has a nonzero K-rational point. In general, this does not imply that K is a splitting field for F since Z need not be K ⊗_{k} L. The assumption that KF_{1} ⊗_{k} N^{0} is irreducible is equivalent, by lemma 3.3, to M̃ := KF_{1} ⊗_{F_{1}} M is irreducible and primitive. One applies lemma 3.4 to Ñ := sym^{2}M̃. Thus sym^{2}Ñ satisfies (1) or (2) of lemma 3.4. Then the same holds for sym^{2}(K ⊗_{k} N^{0}). In particular the quadratic form associated to L has a non-zero K-rational point and thus K is a splitting field for F. □

3.2.1 An example with A_{1} = x and A_{2} = s_{0} − s_{1}x

We keep the notations of subsection 3.2. Assume that s_{0}, s_{1} ∈ Q and s_{0} ≠ 0, s_{1} ≠ 0, 1. The central simple algebra F over Q(x) need not be a skew field. In fact, K ⊃ Q(x) is a splitting field for F if and only if the quadratic equation A_{1}X^{2} + A_{2}Y^{2} − Z^{2} = 0 has a solution (X, Y, Z) ≠ (0, 0, 0) in K^{3}. For K = C_{K}(x) this condition is equivalent to s_{0} and s_{1} being squares in C_{K}. Indeed, if s_{0}, s_{1} are squares, then one easily sees that there are a, b ∈ C_{K} such that xa^{2} +(s_{0} −s_{1}x)b^{2} −1 = 0. On the other hand, let (X, Y, Z) ∈ C_{K}(x)^{3} be a non-trivial solution of the quadratic equation. Then one can normalize this solution such that X, Y, Z ∈ C_{K}[x] and the g.c.d. of X, Y, Z is 1. Substitution of x = 0 and x = \frac{s_{0}}{s_{1}} in the equation xX^{2} + (s_{0} − s_{1}x)Y^{2} − Z^{2} = 0 yields that s_{0} and s_{1} are squares in C_{K}.

Now we suppose that not both s_{0} and s_{1} are squares in Q. Then F is a quaternion field. Examples of splitting fields for F are: Q(\sqrt{s_{0}}, \sqrt{s_{1}})(x) or F_{1} = Q(\sqrt{x}) or Q(\sqrt{s_{0} − s_{1}x + m^{2}x}) for any m ∈ Q. The F -vector space M = F e is made into a differential module over F by ∂e = de and d = b_{1} + xb_{2}. We want to apply 3.6 with K = C_{K}(x) where C_{K} is any algebraic extension of Q. Thus we have to show that Ñ := \overline{Q}(\sqrt{x})⊗_{Q(x)}N^{0} is irreducible, since the composite of the fields \overline{Q}(x) and F_{1} is \overline{Q}(\sqrt{x}). Suppose that M contains a one-dimensional F_{1}-vector space V, invariant under ∂. Then b_{2}V is also a 1-dimensional F_{1}-vector space, invariant under ∂ (indeed, b_{1}b_{2} = −b_{2}b_{1} and b′_{2} = \frac{A′_{2}}{2A_{2}} b_{2}) and M = V ⊕ b_{2}V. Thus M is always semisimple over F_{1}. Then also Ñ is semi-simple. Suppose that Ñ is reducible. Then Ñ has a direct sum decomposition. The Galois group of the extension \overline{Q}(\sqrt{x}) ⊃ \overline{Q}(x) acts on these direct sum decompositions. From this one concludes that \overline{Q}(x) ⊗ N^{0} is also reducible and moreover that \overline{Q}(x) ⊗ N^{0} contains a submodule of dimension 2. Let L_{3} denote the minimal monic operator L_{3} for the cyclic vector b_{2}e ⊗ e of N^{0}. We will prove that Ñ is irreducible by showing that L_{3} has no right hand factor of order 1 in \overline{Q}(x)[∂]. One calculates that L_{3} is equal to

x(s_{0} − s_{1}x)∂^{3} + 3(s_{0}/2 − s_{1}x)∂^{2} + (4(s_{1} − 1)x − 3s_{1}/4 − 4s_{0})∂ + 6(s_{1} − 1).

Suppose that ∂−u with u ∈ \overline{Q}(x) is a right hand factor of L_{3}. We make now a local analysis at the singular points x = 0, s_{0}/s_{1}, ∞ of L_{3}. The first two points are regular singular with local exponents 0, 1, 1/2. The point ∞ is irregular singular and has only one “generalized local exponent” which is unramified, i.e., does not involve a root of the local parameter \frac{1}{x} at ∞. This exponent is 3/2 and gives a local right hand factor of the form ∂ + 3/2x^{−1} + · · ·. Then u has the form u = \frac{l_{0}}{x} + \frac{l_{1}}{x−s_{0}/s_{1}} + \frac{f′}{f} with f ∈ \overline{Q}[x], where l_{0}, l_{1} ∈ \{0, 1, 1/2\}. This cannot produce the prescribed local right hand factor at ∞. Let K = C_{K}(x). From theorem 3.6 one concludes that there exists L_{2} ∈ K[∂] whose symmetric square is equivalent to L_{3}, if and only if the equation A_{1}X^{2} + A_{2}Y^{2} − Z^{2} = 0 has a non-trivial solution in K^{3}. The latter is equivalent to \sqrt{s_{0}}, \sqrt{s_{1}} ∈ C_{K}.

Observation. One can compute a monic operator L_{2} ∈ Q(\sqrt{s_{0}}, \sqrt{s_{1}})(x)[∂] with Sym^{2}L_{2} equivalent to L_{3} above. Suppose that Q(\sqrt{s_{0}}, \sqrt{s_{1}}) has degree 4 over Q. Let σ be a non-trivial automorphism of Q(\sqrt{s_{0}}, \sqrt{s_{1}}). Then σ(L_{2}) is not equivalent to L_{2}. But according to theorem 4.7 it must be projectively equivalent (see section 4) to L_{2}. We verified by computer computation the following: Let σ_{i} interchange \sqrt{s_{i}} and −\sqrt{s_{i}}, and leave \sqrt{s_{j}} invariant where j ≠ i. Let t_{0} = \sqrt{A_{1}A_{2}} and let t_{1} = \sqrt{A_{2}}. Then σ_{0}(L_{2}) is equivalent to L_{2} ⊗(∂ − \frac{t′_{0}}{t_{0}} ) and σ_{1}(L_{2}) is equivalent to L_{2} ⊗ (∂ − \frac{t′_{1}}{t_{1}} ). If one of these operators is equivalent to L_{2} then (the module corresponding to) L_{2} is imprimitive by lemma 4.1. The latter is excluded by the irreducibility of L_{3}.

3.2.2 Quaternion fields with general A_{1} and A_{2}

Consider elements A_{1}, A_{2} ∈ Q(x) with A′_{1} ≠ 0 ≠ A_{2}. Even if F is not a skew field, one can define ∂ on M = F e by ∂e = de where d = b_{1} + xb_{2}. Again M is a differential module over F_{1} and sym_{F_{1}}^{2}M = F_{1} ⊗_{Q(x)} N^{0}. Moreover b_{2}e ⊗ e is a cyclic vector for N^{0}. Let L_{3} denote the monic operator of order 3 with L_{3}(b_{2}e ⊗ e) = 0. In the following we will make the equivalence between L_{3} and the second symmetric power of some operator L_{2} over some field K ⊃ Q(x) explicit. For this purpose we introduce an operator R of order < 3, with coefficients in K ⊃ Q(x). One describes R by: the least common left multiple LCLM(R, L_{3}) of R and L_{3} has the form Sym^{2}L · R = B · L_{3} for some L_{2} ∈ K[∂] of order 2. In other words, R provides the isomorphism K[∂]/K[∂]Sym^{2}L_{2} → K[∂]/K[∂]L_{3} of differential modules.

L_{3} = 4A_{2}∂^{3} − 2(A′_{1}A_{2}/A_{1} + 2A′′_{1}A_{2}/A′_{1} − 2A′_{2})∂^{2}

−(16A_{1}A′_{1}^{2}A_{2} + A′_{2}^{2}/A_{2} + A′_{1}A′_{2}/A_{1} + 2A′′_{1}A′_{2}/A′_{1} − 2A′′_{2} + 16)∂

+8(A′_{2}/A_{2} + A′_{1}/A_{1} + 2A′′_{1}/A′_{1})
R = uA_{2}∂^{2} + (2A′_{1}A_{2}w + A′_{2}u/2)∂ + 4A′_{1}A_{2}v − 4u

where (u, v, w) is a non-zero point on the conic

A_{1}u^{2} + A_{2}v^{2} − w^{2} = 0 (1)

Assuming that K ⊗_{Q(x)} sym^{2}N^{0} has only one submodule of dimension 1, namely the one with generator

\frac{A_{2}}{4} n_{1}^{0} ⊗ n_{1}^{0} − \frac{A_{2}}{4A_{1}} n_{2}^{0} ⊗ n_{2}^{0} − n_{3}^{0} ⊗ n_{3}^{0} ,

we have a one to one correspondence between all non-zero points (u, v, w) ∈ K^{3} on the conic, and all operators R ∈ K[∂] of order < 3 with the required property LCLM(R, L_{3}) = Sym^{2}L_{2} · R for some L_{2} of order 2. We note that \overline{Q}(x) is a C_{1}-field, and hence there is always a field K of the form C_{K}(x) with [C_{K} : Q] < ∞ such that (1) has a non-trivial solution. The degree [C_{K} : Q] can be arbitrarily high as is shown in the following example:

A_{1} = x, A_{2} = (x − 2)(x − 3)(x − 5) ⋯ (x − p_{n}),

where p_{n} is the nth prime number. The smallest field extension C_{K} of Q for which (1) has a non-trivial solution in K = C_{K}(x) is Q(\sqrt{2}, .., \sqrt{p_{n}}, \sqrt{(−1)^{n}}).

Remark: For any specific choice, say of A_{1}, A_{2} ∈ Q[x] with A′_{1} ≠ 0 ≠ A_{2}, we need to verify that L_{3} has the desired properties (i.e., K ⊗_{Q(x)} sym^{2}N^{o} has only one submodule of dimension 1). Suppose that the differential Galois group G of \underline{M} := \overline{Q}F_{1} ⊗_{F_{1}} M satisfies G ⊃ SL_{2}. Then the differential Galois group of \overline{F_{1}} ⊗_{F_{1}} M is G^{o} and contains SL_{2}. Then for any algebraic extension K of Q(x), the second symmetric power of K ⊗_{Q(x)} N^{0} has a unique submodule of dimension 1. If at a point p formal solutions involve logarithms, then the differential Galois group of \underline{M} contains SL_{2}. Indeed, the differential Galois group is a reductive group (recall that \underline{M} is semi-simple) and contains the additive group G_{a}. Examples with this feature are obtained by a different choice for ∂e, namely ∂e = (\frac{A′_{1}}{A_{1}−1}b_{0} + \frac{A′_{1}}{A_{1}(A_{1}−1)}b_{1} + \frac{A′_{1}}{A_{2}} b_{2})e. Suppose that the point p satisfies A_{1}(p) = 1 and A_{2}(p) ≠ 0, then one can verify that local solutions at p contain logarithms (it is sufficient to check this for the case A_{1} = x, A_{2} ∈ \overline{Q(x)} because one can then generalize the result by applying a pullback x ↦ A_{1}). Thus logarithms will appear in local solutions if A_{2} is not a multiple of A_{1}−1.

The operator obtained this way is:

L_{3} = 4∂^{3} + (4B_{2} − 12A′′_{1}/A′_{1} + (6A_{1} − 2)A′_{1}/B_{1})∂^{2}
+(2A′′_{2}/A_{2} + (3A_{1} − 1)A′_{1}B_{2}/B_{1} − 6A′′_{1}B_{2}/A′_{1} − 4A′′′_{1}/A′_{1}

−B_{2}^{2} + 12(A′′_{1}/A′_{1})^{2} − 2(3A_{1} − 1)A′′_{1}/B_{1} − 16A′_{1}^{2}/(A_{1}(A_{1} − 1)^{2})

−16A′_{1}^{2}/A_{2})∂ + 8(A′_{1}^{2}B_{2} − (3A_{1} − 1)A′_{1}^{3}/A_{2})/A_{2}
where B_{1} = A_{1}(A_{1} − 1) and B_{2} = A′_{2}/A_{2}.

This operator has the same conic (1). After, if necessary, replacing A_{1} with c^{2}A_{1} for some non-zero c ∈ Q, we obtain that A_{2} is not a multiple of A_{1} − 1 (if A_{1} ∈ Q then take c ∈ Q(x) instead of in Q). The conic (1) changes into an equivalent one. We conclude that for any algebraic extension K of Q(x), the operator L_{3} is equivalent to a symmetric power of some L_{2} ∈ K[∂] if and only if (1) has a non-zero solution in K^{3}. Furthermore, such examples exist for every non degenerate conic over Q(x).

4 Projective equivalence

Some notations and definitions. Let k be a differential field with field of constants C_{k}. Put \underline{k} = \overline{C_{k}}k, where \overline{C_{k}} is the algebraic closure of C_{k}. The trivial 1-dimensional differential module is denoted by 1. The determinant det(M) of a differential module M is the 1-dimensional module Λ^{n}M, where n is the dimension of M. For a 1-dimensional differential module L, one writes L^{⊗n} for the tensor product L ⊗ ⋯ ⊗ L of n copies of L. Two differential modules M_{1}, M_{2} will be called projectively equivalent if there exists a differential module L of dimension 1 such that M_{2} is isomorphic to L ⊗ M_{1}. Suppose that C_{k} is algebraically closed, then M_{1}, M_{2} correspond to representations of the universal differential Galois group U on finite dimensional C_{k}-vector spaces. These representations are projectively equivalent if and only M_{1} and M_{2} are projectively equivalent. The translation in terms of monic differential operators L_{1}, L_{2} ∈ k[∂] reads as follows: L_{1} and L_{2} are projectively equivalent if there exists f ∈ k such that the k-algebra automorphism ϕ_{f} : k[∂] → k[∂], given by ϕ_{f}(∂) = ∂ + f, has the property that ϕ_{f}(L_{1}) is equivalent to L_{2}.

The problem. Let M be an irreducible differential module over K of dimension n. Suppose that ^{σ}M is projectively equivalent to M for every σ ∈ Gal(K/k). The problem is to find the fields k ⊂ ℓ ⊂ K for which there exists a differential module N, projectively equivalent to M, such that N descends to ℓ. The main case of interest is K = \underline{k} and ℓ = C_{ℓ}k.

Lemma 4.1 Suppose that M is irreducible of dimension n > 1 over k and that k contains a primitive n^{th}-root of unity. The following are equivalent:

(1) There exists a 1-dimensional L ≅ 1̸ with L ⊗ M ≅ M.

(2) There exists a cyclic field extension k ⊂ k(t) with equation t^{d} = f ∈ k of degree d ≠ 1 dividing n, such that [\underline{k}(t) : \underline{k}] = d and k(t) ⊗ M is a direct sum of d irreducible differential submodules over k(t).

Proof. (1)⇒(2). det(M) ≅ det(L⊗M) ≅ L^{⊗n}⊗det(M) and so L^{⊗n} = 1. Let d be the minimal divisor of n such that L^{⊗d} = 1. Put L = ke with ∂e = be. There exists an element f ∈ k^{∗} with \frac{f′}{f} = db. The isomorphism M → M ⊗ke can be written as m ↦ ϕ(m) ⊗ e, where ϕ : M → M is a bijective k-linear map. One finds that ϕ∂ϕ^{−1} = ∂ + b and ϕ^{d}∂ϕ^{−d} = ∂ + db = f^{−1}∂f. Then fϕ^{d} commutes with ∂ and thus fϕ^{d} ∈ C_{k}^{∗}. After changing f we may suppose that fϕ^{d} = 1. One considers the field extension k(t) ⊃ k, given by t^{d} = f. It is easily seen that [\underline{k}(t) : \underline{k}] = d. Define ψ := tϕ : k(t) ⊗ M → k(t) ⊗ M. Then ψ∂ = ∂ψ and ψ^{d} = 1. Let N_{0}, …, N_{d−1} ⊂ N =: k(t) ⊗ M denote the eigenspaces of ψ corresponding to the eigenvalues ζ_{d}^{i}, i = 0, …, d − 1, where ζ_{d} denotes a primitive d^{th}-root of unity. Then N = ⊕_{i}N_{i} and the N_{i} are submodules of N. We will use a cyclic notation for the N_{i}, e.g. N_{d} = N_{0}. Let σ ∈ Gal(k(t)/k) be the generator of this Galois group, given by σ(t) = ζ_{d}^{−1}t. Then σ acts also on N and one has σ(N_{i}) = N_{i+1} for all i. Indeed, for n_{i} ∈ N_{i} one has ψ(σn_{i}) = tϕσn_{i} = tσϕn_{i} = ζ_{d}σtϕn_{i} = ζ_{d}σζ_{d}^{i}n_{i} = ζ_{d}^{i+1}σn_{i}. Now suppose that (say) N_{0} has a non-trivial submodule N′_{0}. Then N′ := N′_{0} ⊕ σN′_{0} ⊕ ⋯ ⊕ σ^{d−1}N′_{0} is a non-trivial submodule of N, stable under the action of σ. Then the set of the σ-invariant elements of N′ forms a non-trivial submodule of M. This contradicts the assumptions.

(2)⇒(1). Write again N = k(t) ⊗ M and let σ have the same meaning as before. Take an irreducible submodule N_{0} of N. Put N_{i} := σ^{i}N_{0} for i = 0, …, d − 1. Then N′ := ∑_{i=0}^{d−1} N_{i} is a differential submodule of N, invariant under the action of σ. The set of the σ-invariant elements of N′ forms a non-zero submodule of M. Hence N′ = N. The assumption in (2) implies that N = ⊕_{i=0}^{d−1}N_{i}. Define the k(t)-linear map ψ : N → N, by ψn_{i} = ζ_{d}^{i}n_{i} for any i = 0, …, d − 1 and any n_{i} ∈ N_{i}. Then ψ∂ = ∂ψ and ψσ = ζ_{d}σψ. Define the k(t)-linear ϕ : N → N by ϕ = t^{−1}ψ. Then ϕ∂ϕ^{−1} = ∂ + \frac{f′}{df} and σϕ = ϕσ. The last equality implies that ϕ(M) = M and that the restriction ϕ′ of ϕ to M is a k-linear bijection. The relation ϕ′∂ = (∂ + \frac{f′}{df})ϕ′ implies that L ⊗ M is isomorphic to M, where L = ke is given by ∂e = \frac{f′}{df} e. The assumption [\underline{k}(t) : \underline{k}] = d implies that d is minimal such that L^{⊗d} ≅ 1. □

Remarks. A differential module M of dimension n > 1 will be called cyclicimprimitive if M satisfies property (1) of lemma 4.1. We recall that an irreducible action of a group G on a finite dimensional vector space V is called imprimitive if V has a direct sum decomposition V = V_{1} ⊕ ⋯ ⊕ V_{d} (with d > 1) such that G permutes the V_{i}. This induces a homomorphism G → S_{d} which has as image a transitive subgroup of S_{d} since the action of G is irreducible. The action of G is called cyclic-imprimitive if the image of G → S_{d} is a cyclic group of order d. If C_{k} is algebraically closed then the solution space V of the irreducible differential module M and the action of the differential Galois group G on V is well defined. In this situation, it is easily seen that M is cyclic-imprimitive if and only if the action of G on V is cyclic-imprimitive. An example of a cyclic-imprimitive operator is given in case 3 in the table in section 1 (if k = C_{k}(x), K = C_{K}(x) then case 3 implies part (1) of lemma 4.1).

Corollary 4.2 Let k = C_{k}(x) and \underline{k} = \overline{C_{k}}(x) and let M_{1}, M_{2} be differential modules over \underline{k}, which descend to differential modules N_{1}, N_{2} over k. Suppose that M_{1} and M_{2} are projectively equivalent and that M_{1} is not cyclic-imprimitive. Then N_{1} and N_{2} are projectively equivalent.

Proof. There is a one-dimensional module L over \underline{k}, unique up to isomorphism because M_{1} does not satisfy 4.1 (1), such that L ⊗ M_{1} ≅ M_{2}. For any σ in the Galois group of \underline{k}/k one has ^{σ}L ⊗ ^{σ}M_{1} ≅ ^{σ}M_{2}. Since ^{σ}M_{i} ≅ M_{i} for i = 1, 2, one has ^{σ}L ≅ L. By 2.8, there exists a one dimensional L_{0} over k with L ≅ \underline{k} ⊗ L_{0}. The two modules L_{0} ⊗ N_{1} and N_{2} are isomorphic after tensoring with \underline{k} over k. By 2.3, L_{0} ⊗ N_{1} is isomorphic to N_{2}. □

Example 4.3 Corollary 4.2 is no longer valid if M_{1} is cyclic-imprimitive. We give two examples. Let M_{i} = \overline{Q}(x)⊗_{Q(x)}N_{i} for i = 1, 2, where N_{1}, N_{2} denote the differential modules over defined by the differential operators L_{1}, L_{2}. In the first example one chooses

L_{1} = ∂^{2} + \frac{3(5x^{4} − 2)}{2x^{2}(x^{4} − 2)^{2}} and L_{2} = ∂^{2} − \frac{3x^{2}(x^{4} − 10)}{4(x^{4} − 2)^{2}}.

One can verify the following. The differential Galois group of M_{1} is D_{2}^{SL_{2}}. Moreover sym^{2}N_{1} ≅ sym^{2}N_{2} and the modules N_{1}, N_{2} are not projectively equivalent. They become projectively equivalent after extending the constants with a solution of the equation t^{4} + 2 = 0 (not t^{4} − 2 as the denominators would suggest). The proof of these statements can be deduced from part (2d) of the proof of theorem 4.7. In the second example one fixes c ∈ Q, not a square, an integer n > 2, and considers monic operators L_{1}, L_{2} ∈ Q(x)[∂] of degree 2, defined by the data: three regular singularities \sqrt{c}, −\sqrt{c}, ∞, exponent-difference 1/2 at ±\sqrt{c} for L_{1}, L_{2}, L_{1} resp. L_{2} have exponent-difference \frac{1}{n} resp. 1 + \frac{1}{n} at x = ∞. The differential Galois group of M_{1} is D_{n}^{SL_{2}}, n > 2. The modules N_{1}, N_{2} are not projectively equivalent, but become projectively equivalent after extending the constants with \sqrt{c}. The proof of these statements can be deduced from part (2b) of the proof of theorem 4.7.

Theorem 4.4 Let K be Galois over k and G = Gal(K/k). Let M be an irreducible, not cyclic-imprimitive, differential module of dimension n > 1 over K such that det(M) descends to k and End_{K[∂]}(M) = C_{K}. Suppose that for any σ ∈ G the twisted differential module ^{σ}M is projectively equivalent to M. Then:

(1) M induces a 2-cocycle class \overline{c} ∈ H^{2}(G, K^{∗}), having an order d with d|n. Let F denote the (skew) field with center k associated to \overline{c}.

(2) A field k ⊂ ℓ ⊂ K is a splitting field for F if and only if M is projectively equivalent to a module N that descends to ℓ.

Proof. (1) For σ ∈ G there is an isomorphism ϕ(σ) : ^{σ}M → M ⊗ L(σ) of differential modules. Then one has isomorphisms:

στM →^{^{σ}ϕ(τ)} ^{σ}(M ⊗ L(τ)) = ^{σ}M ⊗ ^{σ}L(τ) →^{ϕ(σ)⊗1} M ⊗ L(σ) ⊗ ^{σ}L(τ)

and ϕ(στ) : ^{στ}M → M ⊗ L(στ). The assumption on M implies that L(στ) ≅ L(σ) ⊗ ^{σ}L(τ). From det(M) = det(^{σ}M) = det(M) ⊗ L(σ)^{⊗n} one concludes that L(σ)^{⊗n} = 1. Fix a basis e(σ) for each L(σ). Let A(σ) be the σ-linear map M → M for which m ↦ A(σ)(m) ⊗ e(σ) is the σ-linear map M → M ⊗ L(σ) associated to ϕ(σ). Then A(σ)∂ = (∂ + a(σ))A(σ) with a(σ) ∈ K given by ∂e(σ) = a(σ)e(σ). One has c(σ_{1}, σ_{2})A(σ_{1}σ_{2}) = A(σ_{1})A(σ_{2}) with c(σ_{1}, σ_{2}) ∈ K^{∗}. Moreover a(σ_{1}σ_{2}) = a(σ_{1}) + σ_{1}(a(σ_{2})) + \frac{c(σ_{1},σ_{2})′}{c(σ_{1},σ_{2})} . The collection \{c(σ_{1}, σ_{2})\} defines a 2-cocycle with class \overline{c} ∈ H^{2}(G, K^{∗}). The 2-cocycle class \overline{c}^{n} is associated to the differential module det(M). By assumption, det(M) descends to k and thus \overline{c}^{n} = 1.

(2) It suffices to consider the case ℓ = k. Suppose that \overline{c} = 1, then one may suppose that c(σ_{1}, σ_{2}) = 1 for all σ_{1}, σ_{2}. Now {a(σ)} is a 1-cocycle with values in K. Such a 1-cocycle is trivial and thus has the form a(σ) = σ(b)−b for a certain b ∈ K. The differential module N = M ⊗ Ke, with ∂e = be, descends to k. Indeed, for N one has that the corresponding maps A(σ) commute with ∂ and A(σ_{1})A(σ_{2}) = A(σ_{1}σ_{2}) holds for all σ_{1}, σ_{2}. On the other hand, if N with property (2) is given, then clearly \overline{c} = 1. □

The condition that det(M) descends to k is not a serious restriction because every M is projectively equivalent to a module with this property. Note that if K = \overline{k} then L(σ)^{⊗n} = 1 implies L(σ) = 1, so the theorem follows from 2.10 (d) in this case.

Example 4.5 A skew field F of dimension 9 over Q(x). Let α be a root of the polynomial z^{3}−3z−1. Then Q(α) is Galois over Q. Let σ be the automorphism that sends α to 2 − α^{2}. Let F be the skew field with center Q(x), with basis 1, b, b^{2} as Q(α, x)-vector space, and multiplication rules b^{3} = x and bf = σ(f)b.

We turn the vector space F e into a Q(α, x)[∂]-module by defining α′ = 0, b′ = \frac{b}{3x} and ∂e = \frac{α+b}{x}e. Taking a cyclic vector then leads to the following operator:

L = L_{0}L_{1}L_{2} − x where L_{i} = x∂ − σ^{i}(α) − i/3.

Now L, σ(L), σ^{2}(L) are not equivalent but are projectively equivalent. They are not projectively equivalent to an element of Q(x)[∂].

If we increase our differential field to Q(α, x^{1/3}) then these three modules become isomorphic and descend to a module over Q(x^{1/3}) given by the following operator

9x^{3}(x + 1)∂^{3} + 3x^{2}(9 − x^{\frac{1}{3}} + x^{\frac{2}{3}} + 8x)∂^{2} − 3x(6 + 2x^{\frac{1}{3}} − x^{\frac{2}{3}}
+7x + x^{\frac{4}{3}} )∂ − 9 + 5x^{\frac{1}{3}} − 11x^{\frac{2}{3}} − 10x − 3x^{\frac{4}{3}} − 3x^{\frac{5}{3}} − 9x^{2}

This operator was obtained through a cyclic vector computation of F e, but this time viewed as Q(b)[∂]-module.

Remarks 4.6

(1) Let the differential module M over K of dimension n > 1 satisfy the assumptions of theorem 4.4. Then sym_{K}^{n}M is projectively equivalent to a module that descends to k. Indeed, the 2-cocycle associated to this module is \overline{c}^{n} = 1.

(2) If K = \underline{k} and M is of dimension n = 2 over K then a converse for (1) can be obtained from 4.7 below, and one finds: If sym^{2}M is projectively equivalent to a module that descends to k then M is projectively equivalent to its conjugates over k.

(3) We recall the classification of irreducible algebraic subgroups of SL_{2}(C), where C is an algebraically closed field of characteristic 0

D_{∞}^{SL_{2}}, D_{n}^{SL_{2}}, A_{4}^{SL_{2}}, S_{4}^{SL_{2}}, A_{5}^{SL_{2}}, SL_{2}(C).

The first five groups are the pre-images in SL_{2}(C) of the subgroups D_{∞}^{pr}, D_{n}, A_{4}, S_{4} and A_{5} of PSL_{2}(C). Further D_{∞}^{pr} is the projective dihedral group consisting of the elements of PSL_{2}(C) which stabilize the subset {0, ∞} of P^{1}(C). For more details see [Ko]. We will use this classification to prove theorem 4.7.

(4) Theorem 4.7 below does not hold for higher symmetric powers nor for second symmetric powers of modules of dimension > 2. We give three examples where M_{1}, M_{2} are not projectively equivalent, and sym^{i}M_{1} ≅ sym^{i}M_{2} with i = 3 for (a), (b) and i = 2 for (c). Let E(Q) be the one-dimensional module given by ∂e = Q/x e. Then E(Q_{1}) ⊗ E(Q_{2}) ≅ E(Q_{1} + Q_{2}) and E(1) ≅ E(0). (a). M_{1} = E(0) ⊕ E(\frac{1}{5}) and M_{2} = E(\frac{1}{15}) ⊗ (E(0) ⊕ E(\frac{2}{5})). (b). M_{1} = E(0) ⊕ E(\frac{1}{10}) ⊕ E(\frac{3}{10}) and M_{2} = E(\frac{1}{5}) ⊗ (E(0) ⊕ E(\frac{−1}{10}) ⊕ E(\frac{−3}{10})). (c). M_{1} = E(0) ⊕ E(\frac{1}{7}) ⊕ E(\frac{3}{7}) and M_{2} = E(\frac{1}{14}) ⊗ (E(0) ⊕ E(\frac{1}{7}) ⊕ E(\frac{5}{7})).

Theorem 4.7 Let M_{1}, M_{2} denote differential modules over \underline{k} of dimension 2. If sym2 ; \underline{k}M_{1} ≅ sym2 ; \underline{k}M_{2}, then there exists a one-dimensional differential module L over \underline{k} such that M_{2} ≅ L ⊗ M_{1}.

Remark: Corollary 4.2 implies that over rational functions this result is also valid for non-algebraically closed field of constants except (see examples 4.3) in the imprimitive case (cases (2b) and (2d) in the proof below). The proof below is long because it distinguishes many cases. Bas Edixhoven informed us that a shorter proof can be obtained with the following ideas: replace the groups G_{i} by the largest group that stabilizes both quadrics, consider the intersection R of these quadrics as a closed subscheme of length 4, and distinguish cases based on the structure of R.

Proof. The theorem is in fact a statement concerning representations. The translation is as follows. Let U denote the universal differential Galois group of the field \underline{k}. The category of the differential modules over \underline{k} is equivalent to the category of the finite dimensional C-linear representations of the affine group scheme U. Here C = C\underline{k} is the field of constants of \underline{k}. One associates to a differential module M over \underline{k} its solution space V equipped with the action of U. Let ρ_{i} : U → GL(V_{i}) for i = 1, 2, denote the representations associated to M_{i}. Put W_{i} = sym^{2}V_{i} equipped with the induced representation sym^{2}ρ_{i}. There is given an isomorphism B : W_{1} → W_{2} between the two representations. Now it suffices to show that there exists a C-linear bijection A : V_{1} → V_{2} such that sym^{2}A = B. Indeed, for any g ∈ U one has that Aρ_{1}(g)A^{−1} and ρ_{2}(g) have the same second symmetric power. Hence there exists a χ(g) ∈ {±1} such that Aρ_{1}(g)A^{−1} = χ(g)ρ_{2}(g). The map g ∈ U ↦ χ(g) ∈ {±1} is a one dimensional representation and corresponds to a one-dimensional differential module L over \underline{k}. It follows that M_{1} ≅ L ⊗ M_{2}.

Since we are dealing with only two differential modules M_{1} and M_{2}, we may replace in the sequel the (somewhat fancy) affine group scheme U by the differential Galois group of M_{1} ⊕ M_{2}, which is an ordinary linear algebraic group over C.

As before, one considers the canonical surjective map sym_{C}^{2}W_{i} → sym^{4}V_{i} and its one-dimensional kernel K_{i} for i = 1, 2. One observes that a C-linear bijection B : W_{1} → W_{2} has the form sym^{2}A for some C-linear bijection A : V_{1} → V_{2} if and only if sym^{2}B : sym_{C}^{2}W_{1} → sym_{C}^{2}W_{2} maps K_{1} to K_{2}.

Let G_{i} denote the image of ρ_{i}. The isomorphism between W_{1} and W_{2} implies that the induced morphisms U → G_{i}/{±1} ∩ G_{i}, i = 1, 2 coincide.

(1a) Suppose that V_{1} is reducible and that it has precisely one proper invariant subspace L_{1}. It easily follows that the only non-trivial invariant subspaces of W_{1} are L_{1} ⊗ L_{1} and L_{1} ⊗ V_{1}. Also V_{2} has a unique proper invariant subspace L_{2}, since G_{1}/{±1} ≅ G_{2}/{±1}. Thus L_{1} ⊗V_{1} is isomorphic to L_{2} ⊗ V_{2}. Hence V_{2} ≅ L_{1} ⊗ L_{2}^{−1} ⊗ V_{1}.

(1b) Suppose that V_{1} has precisely two proper invariant subspaces. We may assume that the representation V_{1} is the sum of the trivial character (denoted by 1) and a non-trivial character χ_{1}. Because G_{1}/{±1}∩G_{1} ≅ G_{2}/{±1}∩G_{2} one has that V_{2} is the direct sum of two distinct characters χ_{2}, χ_{3}. The two sequences of characters 1, χ_{1}, χ_{1}^{2} and χ_{2}^{2}, χ_{3}^{2}, χ_{2}χ_{3} are equal up to their order. Suppose that χ_{2}^{2} ≠ 1 ≠ χ_{3}^{2}. Then we may suppose that χ_{2}χ_{3} = 1, χ_{2}^{2} = χ_{1}, χ_{3}^{2} = χ_{1}^{2}. Then χ_{1}^{3} = 1, χ_{2} = χ_{1}^{2}, χ_{3} = χ_{1} and V_{2} = χ_{1} ⊗ V_{1}. If say χ_{3}^{2} = 1, then after changing V_{2} into χ_{3} ⊗ V_{2}. Thus we may suppose that V_{2} = 1 ⊕ χ_{2}. If χ_{1} = χ_{2}, then V_{1} ≅ V_{2}. If χ_{1} ≠ χ_{2}, then χ_{1} = χ_{2}^{2}, χ_{1}^{2} = χ_{2} and χ_{1}^{3} = 1. Thus V_{1} = 1 ⊕ χ_{1} = χ_{1} ⊗ (1 ⊕ χ_{2}) = χ_{1} ⊗ V_{2}.

(1c) Suppose that V_{1} has more than two invariant subspaces of dimension 1. Then G_{1}/{±1} = {1}. Also G_{2}/{±1} = {1}. Hence V_{1} and V_{2} differ by a character.

(2) Now we consider the case where G_{1} is irreducible. After multiplying ρ_{1} by a character, one may assume that G_{1} ⊂ SL_{2}. The isomorphism between W_{1} and W_{2} implies that the image G_{2} of the second representation lies in {Z ∈ GL_{2}| det(Z)^{3} = 1}. Using the above notations, we will show that there is a choice of the isomorphism B between W_{1} and W_{2} such that sym^{2}(B) maps K_{1} to K_{2}.

(2a) If G_{1} ∈ {S_{4}^{SL_{2}}, A_{5}^{SL_{2}}, SL_{2}(C)}, then sym^{4}V_{i}, i = 1, 2 has no invariant one-dimensional subspace. Then sym^{2}W_{i}, i = 1, 2 has only K_{i} as invariant subspace of dimension 1. Hence B(K_{1}) = K_{2} and we are done.

(2b) If G_{1} = D_{∞}^{SL_{2}}, then we have to make a more detailed calculation. Let {v_{1}, v_{2}} be a basis of V_{1} such that G_{1} consists of the transformations with determinant 1 which permute the two lines Cv_{1}, Cv_{2} in V_{1}. Then b_{1} = v_{1} ⊗ v_{1}, b_{2} = v_{2} ⊗ v_{2}, b_{3} = v_{1} ⊗ v_{2} is a basis of W_{1} = sym_{C}^{2}V_{1}. The space W_{1} is the direct sum of the irreducible representations Cb_{1} ⊕ Cb_{2} and Cb_{3}. The space sym_{C}^{2}W_{1} is the sum of the irreducible representations Cb_{1} ⊗ b_{1} + Cb_{2} ⊗ b_{2}, Cb_{1} ⊗ b_{2}, Cb_{1} ⊗ b_{3} + Cb_{2} ⊗ b_{3} and Cb_{3} ⊗ b_{3}. The family of all invariant lines in sym_{C}^{2}W_{1} is C(λb_{1} ⊗ b_{2} + μb_{3} ⊗ b_{3}) with λ, μ ∈ C, not both zero. Each member of the family defines a quadric in W_{1}. The family has two special members namely Cb_{1} ⊗ b_{2} and Cb_{3} ⊗ b_{3}, where the quadric consists of two lines or a double line. The situation is similar for W_{2} and sym^{2}W_{2} (with adapted notations v′_{1}, v′_{2}, b′_{1}, b′_{2}, b′_{3}). Thus the isomorphism B : W_{1} → W_{2} sends the family of quadrics of W_{1} to the one of W_{2}. The special quadrics are mapped to the special quadrics and one concludes that Bb_{3} is a multiple of b′_{3} and {Bb_{1}, Bb_{2}} are multiples of {b′_{1}, b′_{2}}. One has the freedom to change the isomorphism B by prescribing Bb_{3} = λb′_{3} for any λ ∈ C^{∗}. In this way one obtains that sym^{2}B maps the line K_{1} with generator b_{1} ⊗ b_{2} − b_{3} ⊗ b_{3} to the line K_{2} with generator b′_{1} ⊗ b′_{2} − b′_{3} ⊗ b′_{3}. Thus the changed B is a sym^{2}A. For G_{1} = D_{n}^{SL_{2}} with n > 2 the same proof works. Note that sym^{2}B depends quadratically on λ, which explains why \sqrt{c} is needed in example 4.3.

(2c) Assume now that G_{1} = A_{4}^{SL_{2}}. Then W_{1} is the irreducible representation D of G_{1}/{±1} = A_{4} of dimension 3. Further sym^{2}W_{1} is the direct sum of three invariant lines L_{1}(0), L_{1}(1), L_{1}(2) and D. The group G_{1}/{±1} = A_{4} acts trivially on L_{1}(0) which is the kernel K_{1} of sym^{2}W_{1} → sym^{4}V_{1}. The action of G_{1} on the other two lines is given by the two non-trivial characters of A_{4}. The space sym^{2}W_{2} has a similar decomposition L_{2}(0)⊕L_{2}(1)⊕L_{2}(2)⊕ D. The notation is chosen such that L_{2}(0) is the kernel K_{2} of sym^{2}W_{2} → sym^{4}V_{2}. Again, L_{2}(0), L_{2}(1), L_{2}(2) correspond to the three 1-dimensional characters of G_{2}/{±1} = A_{4}. However, L_{2}(0) need not correspond to the trivial character of A_{4} and B need not satisfy sym^{2}(B)K_{1} = K_{2}.

Now we replace V_{2} by V′_{2} := L_{2}(0)^{⊗2}⊗V_{2}. This new representation has the same kernel as ρ_{2}. Put W′_{2} = sym^{2}V′_{2} = L_{2}(0)^{⊗4} ⊗ W_{2}. This representation has the same kernel as sym^{2}ρ_{2}. The image of sym^{2}ρ_{2} is identified with A_{4}. As representations of A_{4} the two objects L_{2}(0)^{⊗4} ⊗ D and D are isomorphic, since there is only one irreducible representation of A_{4} with dimension 3. Thus sym^{2}V_{2} and sym^{2}V′_{2} are isomorphic. Then sym^{2}V_{1} and sym^{2}V′_{2} are isomorphic. Let B′ denote the isomorphism. The decomposition of sym^{2}W′_{2} is

L_{2}(0)^{⊗9} ⊕ (L_{2}(1) ⊗ L_{2}(0)^{⊗8}) ⊕ (L_{2}(2) ⊗ L_{2}(0)^{⊗8}) ⊕ D,

where L_{2}(0)^{⊗9} is the kernel K′_{2} of sym^{2}W′_{2} → sym^{4}V′_{2}. Now, as required, sym^{2}(B′)K_{1} = K′_{2} and we conclude that V_{1} and V′_{2} differ by a character. Hence V_{1} and V_{2} differ by a character.

(2d) Assume that G_{1} = D_{2}^{SL_{2}}. Consider a two-dimensional vector space V and a representation ρ : U → SL(V) with image D_{2}^{SL_{2}}. For a suitable basis v_{1}, v_{2} of V the space W = sym^{2}V is a direct sum of the three invariant lines L_{1}, L_{2}, L_{3} with generators e_{1} := v_{1} ⊗ v_{1} + v_{2} ⊗ v_{2}, e_{2} := v_{1} ⊗ v_{1} − v_{2} ⊗ v_{2} and e_{3} := v_{1} ⊗ v_{2}. They correspond to the three characters of D_{2} having order two. Then sym^{2}W is a direct sum of the 3-dimensional space T spanned by e_{i} ⊗ e_{i}, i = 1, 2, 3, having trivial D_{2}-action, and the three lines L_{1} ⊗ L_{2}, L_{1} ⊗ L_{3}, L_{2} ⊗ L_{3}. These lines correspond again to the three characters of D_{2} of order two. The line K, kernel of sym^{2}W → sym^{4}V, lies in T and is generated by an element ∑_{i=1}^{3} c_{i}e_{i}⊗e_{i} with non-zero c_{1}, c_{2}, c_{3} ∈ C. For any other element ∑_{i=1}^{3} d_{i}e_{i}⊗e_{i} in T with all d_{i} ≠ 0, there is an automorphism D of the representation W such that sym^{2}D maps the given element ∑_{i=1}^{3} c_{i}e_{i} ⊗ e_{i} to ∑_{i=1}^{3} d_{i}e_{i} ⊗ e_{i}. Indeed, take D of the form De_{i} = λ_{i}e_{i} for i = 1, 2, 3 where λ_{i}^{2} = d_{i}/c_{i}. Consider, as before, two representations (V_{i}, ρ_{i}) of U having dimension 2, such that the image G_{1} of ρ_{1} is D_{2}^{SL_{2}}. Let an isomorphism B : W_{1} → W_{2} be given. Using the automorphism D above, one changes B into B′ such that sym^{2}B′ maps K_{1} to K_{2}. In the first example of 4.3, the three lines L_{1}, L_{2}, L_{3} are defined over the field of constants Q(\sqrt{2}). The above proof then shows that N_{1}, N_{2} must become projectively equivalent if we extend the constants to C = Q(\sqrt{2}, \frac{λ_{1}}{λ_{3}} , \frac{λ_{2}}{λ_{3}} ). In the example, C turns out to be the splitting field of x^{4}+2, which has degree 8 over Q. In general, if N_{1}, N_{2} are differential modules over Q(x) that become projectively equivalent over \overline{Q}(x), and have differential Galois group D_{n}^{SL_{2}}, then they become projectively equivalent over C′(x) for some field extension C′ of Q of degree ≤ 4 when n = 2, and degree ≤ 2 when n > 2. In the example one can show that the two subfields of C that contain a solution of x^{4} + 2 = 0 are the smallest fields of constants over which N_{1}, N_{2} become projectively equivalent. □

The following explains the constructions with quaternions in section 3.

Corollary 4.8 Suppose that N is an irreducible differential module over \underline{k} of dimension 3, which descends to k. Assume that sym^{2}N has a 1-dimensional submodule, such that the corresponding quadratic form has a \underline{k}-rational point. Then:

(1) There exists a differential module M over \underline{k} with sym^{2}M ≅ N. For every σ ∈ Gal(\underline{k}/k) the modules ^{σ}M and M are projectively equivalent.

(2) Let \overline{c} ∈ H^{2}(Gal(\underline{k}/k), \underline{k}^{∗}) denote the 2-cocycle of order 1 or 2, associated to M. Let F be the quaternion algebra over k associated to \overline{c}. An algebraic extension Ck̃ ⊃ C_{k} yields a splitting field k̃ = Ck̃k for F if and only if there exists a differential module M̃ over k̃ such that \underline{k} ⊗k̃ sym2 ; k̃M̃ is isomorphic to N.

Proof. Apply 3.2, 3.1, 4.7 and 4.4 (observe that the differential Galois group of M is irreducible and primitive since N is irreducible). □

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