Equivalence of differential equations of order one

Accepted Manuscript Equivalence of differential equations of order one L.X.Chau Ngo, K.A. Nguyen, M. van der Put, J. Top

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S0747-7171(14)00109-6 10.1016/j.jsc.2014.09.041 YJSCO 1565

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Journal of Symbolic Computation

Received date: 3 June 2014 Accepted date: 23 September 2014

Please cite this article in press as: Ngo, L.X.C., et al. Equivalence of differential equations of order one. J. Symb. Comput. (2015), http://dx.doi.org/10.1016/j.jsc.2014.09.041

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Equivalence of differential equations of order one L. X. Chau Ngo Quy Nhon University, Department of Mathematics, 170 An Duong Vuong Street, Quy Nhon City, Binh Dinh Province,Vietnam

K. A. Nguyen HCMC University of Technology (HUTECH), Department of Computer Science, 144/24 Dien Bien Phu Str., Ward 25, Binh Thanh Dist., Ho Chi Minh City, Vietnam

M. van der Put Johann Bernoulli Institute, University of Groningen, P.O.Box 407, 9700 AG Groningen, the Netherlands

J. Top Johann Bernoulli Institute, University of Groningen, P.O.Box 407, 9700 AG Groningen, the Netherlands

Abstract The notion of strict equivalence for order one differential equations of the form f (y  , y, z) = 0 with coefficients in a finite extension K of C(z) is introduced. The equation gives rise to a curve X over K and a derivation D on its function field K(X). Procedures are described for testing strict equivalence, strict equivalence to an autonomous equation, computing algebraic solutions and verifying the Painlev´e property. These procedures use known algorithms for isomorphisms of curves over an algebraically closed field of characteristic zero, the Risch algorithm and computation of algebraic solutions. The most involved cases concern curves X of genus 0 or 1. This paper complements work of M. Matsuda and of G. Muntingh & M. van der Put. Keywords: Ordinary differential equations, algebraic curves, local behavior of solutions, normal forms, Painlev´e property, algebraic solutions 2010 MSC: 34M15, 34M35, 34M55

1. Introduction and summary Let K denote a finite extension of C(z) equipped with the C-linear derivation  given ∂f mod (f ) is by z  = 1. Let f ∈ K[S, T ] be absolutely irreducible and assume that d := ∂S Email addresses: [email protected] (L. X. Chau Ngo), [email protected] (K. A. Nguyen), [email protected] (M. van der Put), [email protected] (J. Top) Preprint submitted to Elsevier

September 26, 2014

non zero. To the first order algebraic differential equation f (y  , y) = 0 over K we associate the differential algebra R := K[s, t, d1 ] := K[S, T ]/(f )[ d1 ], its field of fractions K(s, t), and the pair (X, D), where X is the smooth projective algebraic curve over K with function field K(s, t) and D is the derivation on K(s, t) defined by D(z) = 1, D(t) = s. The genus of X will also be called the genus of f . By a solution of f we mean a K-linear differential homomorphism φ : R → F, where F is a differential extension of K such that the field of constants of F is C. More concretely, we may take for F a finite extension of the field of meromorphic functions on some open connected subset of the Riemann surface of K. Equivalently, a possible F could be the field of the convergent Laurent series C({v 1/m }), where v is a local parameter of a point of the Riemann surface of K and m is a positive integer. Two first order algebraic differential equations f1 , f2 over K, inducing pairs (X1 , D1 ) and (X2 , D2 ) are called strictly equivalent if there exists a finite extension L ⊃ K such that L ×K (X1 , D1 ) ∼ = L ×K (X2 , D2 ). If f1 , f2 are strictly equivalent, then for any solution of f1 we obtain finitely many solutions of f2 and visa versa. In the sequel we will rely on algorithms for testing and producing explicit answers to the following questions, briefly discussed in the Appendix. We note that such algorithms exist for the questions (Q1)–(Q4). This does not seem to be the case for question (Q5). Q1. Are two given curves X1 and X2 (over K or over C) isomorphic? Q2. Given is a curve X over K. Does there exists a curve X0 over C with K ×K X ∼ = K ×C X 0 ? Q3. Does the differential equation u = a0 + a1 u + a2 u2 with all a∗ ∈ K have solutions ∗ in K? A well known special case is deciding whether u = a1 u has a solution in K , see [Ri] and [B-D, § 6]. Q4. Find a basis of H 0 (X, L) for a given line bundle L on a given curve X. Q5. Let E be an elliptic curve over C and let K be a finite extension of C(z). Find a basis of the Lang–N´eron group (relative Mordell–Weil group) Q ⊗Z E(K)/E(C). We will sketch procedures for strict equivalence of equations, for strict equivalence to an autonomous equation and for having many algebraic solutions. The algebraic treatment of the Painlev´e property and related problems for first order equations in the work of M. Matsuda and G. Muntingh & M. van der Put is complemented with an algorithmic approach. Finally, the autonomous equations of any genus are classified. 2. The Painlev´ e property and algebraic solutions We recall that a differential equation f (y  , y) = 0 over a finite extension K of C(z) is said to have the Painlev´e property (PP) if the only ‘moving singularities’ for solutions on the Riemann surface of K are poles. The next theorem provides the link between PP and the definition of ‘no moving singularities’ in [M]. Theorem 2.1 ([Mun-vdP], Proposition 4.2). Let (X, D) denote the pair associated to a first order equation f over K. Then f has PP if and only if for every closed point x ∈ X, the local ring OX,x is invariant under D. Remarks 2.2. (1). There exists an algorithm verifying PP for any f . Indeed, the differential algebra R = K[s, t, d1 ] is the coordinate ring of an open affine subset of X. 2

Since R is invariant under D one has to investigate the property D(OX,x ) ⊂ OX,x for the finitely many closed points x ∈ X outside this open subset. (2). Let L ⊃ K be a finite extension. Then f over K has PP if and only if f over L has PP.  Theorem 2.3 ([M, Mun-vdP]). Let (X, D) denote the pair associated to f over K having PP. There are, after replacing K by a finite extension, three cases for (K(X), D) (where K(X) denotes the function field of X), namely d ) with a0 , a1 , a2 ∈ K, not all zero. (i). (K(u), (a0 + a1 u + a2 u2 ) du d 2 (ii). (K(x, y), h · y dx ) with y = x3 + ax + b, a, b ∈ C a non singular elliptic curve and h ∈ K ∗. (iii). (K(X0 ), D0 ) where X0 is a smooth irreducible projective curve over C and D0 is zero on its function field C(X0 ) (which is a subfield of K(X0 )). Remarks 2.4. (1). In [M], the base field K is more general than a finite extension of C(z) and the three cases of (X, D) are given the names Riccati fields, Poincar´e fields and Clairaut fields. (2). The proof of part (iii) in [Mun-vdP] is not constructive. We present in Corollary 2.5 a constructive proof, using ideas from M. Matsuda, in particular his proof of [M, Thm. 14]. Corollary 2.5. Suppose that (X, D) satisfies PP and X is a curve of genus ≥ 2. Let L ⊃ K be a finite extension of K such that all the Weierstrass points of L ×K X are rational over L. Then (L ×K X, D) ∼ = (L × X0 , D0 ), where X0 is a smooth, irreducible, projective curve over C and the derivation D0 of L(X0 ) is zero on the subfield C(X0 ). Proof. We assume that the Weierstrass points of X are rational over K. First, we consider a hyperelliptic curve X and repeat here (for the convenience of the reader) a part of the proof of Proposition 4.3 of [Mun-vdP]. The curve X can be presented by the affine equation y 2 = Q with Q = x(x − 1)(x − a1 ) · · · (x − am ) where m > 1 is odd and 0, 1, a1 , . . . , am are distinct elements of K. Then D(x) = A + By with A, B ∈ K[x]. Thecompletion of the local ring at ∞ can be written as K[[u]] with x−1 = u2 and yum+2 = (1 − u2 )(1 − u2 a1 ) · · · (1 − u2 am ). A small computation shows that D(u) ∈ K[[u]] implies that B = 0 and that A has degree at most one. The condition D(y) ∈ K[x, y] implies 2yD(y) = D(Q) ∈ yK[x, y] ∩ K[x] = QK[x].  Now DQ Q ∈ K[x] implies that A = 0 and all ai = 0. Thus D(x) = D(y) = 0 and Q ∈ C[x]. This finishes this elementary case. We suppose now that X is not hyperelliptic. The canonical embedding of X is given ⊗m 0 by the graded K-algebra R := ⊕ m≥0 H (X, Ω X ). The derivation D on K(X) extends to the sheaf ΩX by the formula D( i fi dgi ) = i D(fi )dgi + fi dD(gi ). Because of PP, the K-vector space of the holomorphic forms H 0 (X, ΩX ) is invariant under D. Likewise D ex⊗m 0 0 tends to the sheaf Ω⊗m X and the K-vector spaces H (X, ΩX ). We claim that H (X, ΩX ) 0 is a trivial differential module, i.e., the canonical map K ⊗C (ker(D, H (X, ΩX )) → H 0 (X, ΩX ) is a bijection. The differential modules H 0 (X, Ω⊗m X ) are images of symmetric powers of H 0 (X, ΩX ) and therefore also trivial. Let R0 := ker(D, R). This is a graded algebra and the canonical map K ⊗ R0 → R is bijective. Now R0 defines a canonical embedded curve X0 over C and X ∼ = K ⊗C X0 and D is zero on C(X0 ). 3

Proof of the claim. (a) Let x denote a Weierstrass point and let π be a generator of the maximal ideal of OX,x . Then D(π) ∈ πOX,x . Proof. Let [x] denote the point x ∈ X considered as a divisor. Also, L(n[x]) denotes the K-vector space of the elements f of K(X) with divisor div(f ) ≥ −n[x]. Since x is a Weierstrass point there exists n ≥ 1 such that L((n − 1)[x]) = L(n[x]) = L((n + 1)[x]). Take f ∈ L(n[x]) \ L((n − 1)[x]). The completion of the local ring OX,x has the form K[[π]] and f = aπ −n + · · · with a ∈ K ∗ . Now D(f ) can only have a pole at x because of PP. Further D(f ) = (a π −n + · · · ) + (−naπ −n−1 + · · · )D(π) and D(f ) ∈ L((n + 1)[x]).  Since L(n[x]) = L((n + 1)[x]) we have that D(π) ∈ πOX,x . (b) H 0 (X, Ω) has a basis ω 1 , . . . , ω g such that D(ω j ) = 0 for all j. Proof. We fix a Weierstrass point x and consider a basis ω1 , . . . , ωg of H 0 (X, Ω) such that ordx (ω1 ) < ordx (ω2 ) < · · · < ordx (ωg ). Let x = x1 , . . . , xr be the set of all Weierstrass points and let πj be a local parameter at xj . Since D(πj ) ∈ πj OX,xj , there is an action of D on ΩX,xj /(πj ) ∼ = K and this action coincides with the differentiation on K. Step by step we will change the basis {ωj } such that at the end D(ω j ) = 0 holds for all j. Step 1. D(ωg ) = aωg for some a ∈ K. There exists a j such that the image b of ωg in ΩX,xj /(πj ) = K is non zero, since r > 2g + 2. Then b = ab holds for some b ∈ K ∗ . Now ω g := 1b ωg satisfies D(ω g ) = 0. Step 2. D(ωg−1 ) = aωg−1 + bω g for some a, b ∈ K. If b = 0, then we replace ωg−1 by b−1 ωg−1 − zω g and arrive at a new ωg−1 satisfying D(ωg−1 ) = aωg−1 for some (new) a ∈ K. As in Step 1, there exists b ∈ K ∗ such that b = ab. Now ω g−1 := 1b ωg−1 satisfies D(ω g−1 ) = 0. Step 3. D(ωg−2 ) = aωg−2 + bω g−1 + cω g for some a, b, c ∈ K is ‘normalized’ in a similar way. By the method of Step 2 one makes b zero. Applying the method again one makes c zero. Then D(ωg−2 ) = aωg−2 for some (new) a ∈ K. As in Step 1, there exists b ∈ K ∗ with b = ab and ω g−2 := 1b ωg−2 satisfies D(ω g−2 ) = 0. Induction finishes the proof.  Algorithm 2.6. Given is a differential equation f = 0 and the corresponding pair (K(X), D). Testing PP for f = 0 is done by Remark 2.2. For f = 0 corresponding to (X, D) having PP we indicate how 2.3 and 2.5 can be made explicit. First one computes the genus g of X. If g = 0, then K(X) = K(u) since K is a C1 -field (see Van Hoeij’s paper [vH] for an explicit computation of u). Then D(u) has the form a0 +a1 u+a2 u2 because of PP. If not all a∗ are zero, then we have arrived at case (i). One may extend the algorithm by computing a ‘standard form’ for D, as in the algorithm presented in §3. For g = 1, one has to produce a rational point of X after enlarging K. A standard computation using this point and possibly a further finite extension of K, yields the affine equation y 2 = x(x − 1)(x − λ) with λ ∈ K ∗ , λ = 0, 1. It follows from PP that λ ∈ C. Let D(x) = h · y with h ∈ K. If h = 0, then we have arrived at (ii). If h = 0, then we are in case (iii). 4

There are algorithms computing Weierstrass points for the case g ≥ 2. One replaces K by a finite extension such that all Weierstrass points are rational. From the number (and/or nature) of Weierstrass points one can see whether the curve is hyperelliptic or not. For a hyperelliptic curve X one can follow the procedure as for elliptic curves. For g > 2 and a non hyperelliptic curve X one applies Coates’ algorithm to get a basis of H 0 (X, Ω). After choosing a Weierstrass point x one can transform this into a basis ω1 , . . . , ωg such that ordx (ω1 ) < · · · < ordx (ωg ). Then one applies the proof of 2.5 and finds a basis {ω j } with all D(ω j ) = 0 of H 0 (X, Ω). Using this basis one finds that the canonical embedding X → P roj(H 0 (X, Ω)) induces a curve X0 over C and an isomorphism X ∼ = K ×C X0 . We note that an affine open subset of X0 has coordinate ωg 2 , . . . , ring C[ ω ω1 ω 1 ]. As we know (and show in Theorem 2.8), all solutions of f = 0 are algebraic for the case that we are considering. These solutions can be expressed in terms ωg 2 of C(X0 ), the field of fractions of C[ ω ω 1 , . . . , ω 1 ]. Theorem 2.7. The following properties of f over K, corresponding to (X, D), are equivalent. (1). All solutions of f are algebraic and lie in a fixed finite extension of K. (2). There exists a finite extension L ⊃ K such that L ×K (X, D) is isomorphic to L ×C (X0 , D0 ) where X0 is a smooth, irreducible, projective curve over C and D0 is the zero derivation on the function field of X0 over C. Proof. (1)⇒(2). Then f has PP since the only singularities, apart from moving poles, are the fixed branch points of the extension L ⊃ K. In case (i) of Theorem 2.3, K(X) = K(u) with u = a2 u2 + a1 u + a0 and there are at least three algebraic solutions α∗ , ∗ = 0, 1, ∞ of u = a2 u2 +a1 u+a0 . The automorphism A, given by α∗ → ∗ for ∗ = 0, 1, ∞ has the property v := Au satisfies D(v) = 0. Thus (2) holds in this case with L = K(α0 , α1 , α∞ ). In case (ii) of Theorem 2.3, one has K(X) = K(x, y) with y 2 = x3 + ax + b, a, b ∈ C, ∂ ∂ + h · y ∂x for some h ∈ K. Let x0 , y0 ∈ L, where L is a finite extension of D = ∂z 2 K satisfy y0 = x30 + ax0 + b and x0 = hy0 . Define elements x1 , y1 ∈ L(X) by the identity (x, y) = (x0 , y0 ) ⊕ (x1 , y1 ), where ⊕ is the addition on the elliptic curve X. Then L(X) = L(x1 , y1 ) with y12 = x31 + ax1 + b. Let A denote the translation on the elliptic x ∂ ∂ ∂ curve over the element (x0 , y0 ). A computation shows that A ∂z A−1 = ∂z + y00 · y ∂x and ∂ ∂ −1 Ay ∂x A = y ∂x . It follows that D(x1 ) = D(y1 ) = 0. In case (iii) of Theorem 2.3, the explicit proof of Corollary 2.5 produces the required result and L ⊃ K is the smallest field such that all Weierstrass points are rational over L. (2)⇒(1) is easily seen. Theorem 2.8. All solutions of f are algebraic if and only if the kernel of D on the function field of X is greater than C. Proof. Let f correspond to the differential algebra R = K[s, t, d1 ] and the pair (X, D). Suppose that D(u) = 0 for some u ∈ K(X), u ∈ C. Write u = ab with a, b ∈ R, b = 0. Let φ : R → F be a solution. If φ(b) = 0, then φ is an algebraic solution since R/bR has finite dimension over K. If φ(b) = 0, then φ(a) − cφ(b) = 0 for some c ∈ C, because ( ab ) = 0. Hence φ is again an algebraic solution. 5

On the other hand, suppose that the differential field (K(X), D) has constants C. Then the inclusion φ : R → K(X) is a transcendental solution. Example 2.9. The equation (y  )a = y b z c with gcd(a, b) = 1, 0 = a = b and c = −a has a a+c a + λ) a−b with λ ∈ C. The equation a full set of algebraic solutions, namely y = ( a−b a+c z c has genus zero and its function field can be written as K(t) with y = ta , y  = tb z a and a+c c a satisfies s = 0. The following t = a1 tb−a+1 z a for suitable K. Then s := ta−b − a−b a+c z are equivalent: (a) the equation has PP, (b) |a − b| = 1, (c) all solutions are in a fixed finite extension of K. Corollary 2.10. Let the autonomous equation f correspond to the pair (X0 , D0 ) where X0 is a curve over C and D0 = 0 is a derivation of the function field of X0 . Then all solutions of f are algebraic over C(z) if and only if there exists a T ∈ C(X0 ) with D0 (T ) = 1. Proof. The equation f defines a differential algebra R0 := C[s, t, d1 ] over C. The kernel of a (non constant) solution φ : R0 → K with [K : C(z)] < ∞ is zero, since D0 = 0. Thus φ extends to an embedding φ : C(X0 ) ⊂ K. The differential equation D0 (T ) = 1 has an algebraic solution, say z, in K. The differential Galois group of the equation D(T ) = 1 is a finite subgroup of the additive group Ga and hence z ∈ C(X0 ). On the other hand, suppose that there exists a T ∈ C(X0 ) with D(T ) = 1. Then any solution φ satisfies φ(T ) = z + c for some c ∈ C. The solution φ is algebraic since [C(X0 ) : C(T )] < ∞. Algorithm 2.11. Solving D(T ) = 1 for the autonomous equation f (y  , y) = 0. Let the pair (X, D) be induced by f . According to Corollary 2.10, it suffices to produce an algorithm for finding a solution of D(T ) = 1 with T ∈ C(X). Consider a closed point X,x = C[[p]]. x ∈ X with local parameter p. Then O A local solution at x has the form T = ak pk + ak+1 pk+1 + · · · ∈ C((p)) and D(T ) = (kak pk−1 + · · · )D(p) = 1. If D(p) has no pole or zero, then T = a0 + a1 p + · · · with a1 = 0. If D(p) has a zero, D(p) = bk pk + · · · , bk = 0, k ≥ 1, then k = 1 is not possible and for k > 1 one has T has a pole of order k − 1. If D(p) has a pole of order −k, then T has a zero of order k + 1. It follows that a possible T with D(T ) = 1 lies in H 0 (X, L) for a known line bundle L. The Coates algorithm [Co] produces a basis β of H 0 (X, L). Testing D(T ) = 1 now reduces to expressing 1 as a C linear combination of D(β). Remarks 2.12. (1). Corollary 2.10 is essentially present in [A-C-F-G] and 2.11 improves on the algorithm proposed in that paper. (2). From the text one can deduce an algorithmic version of Theorem 2.7(1). However, we have not been able to do this for Theorem 2.8, i.e., we do not know how to verify ker(D, K(X)) = C. (3). Suppose that the equation f has PP and the genus of f is zero, then computing algebraic solutions amounts to computing algebraic solutions of a Riccati equation u = a2 u2 + a1 u + a0 . There are algorithms for this problem. In contrast to the above, we do not know whether a simple equation like y  = y 3 + z over C(z) has an algebraic solution. The local solutions are: 6

For z = a = ∞ there is a holomorphic solution y ∈ C{z − a}, depending on the initial  value y(a). Moreover there is a ramified meromorphic solution y = n≥−1 an (z − a)n/2 in C({(z − a)}), depending on a−1 and a2−1 = −1 2 . 2 3 3 For z = ∞ and with t := z1 the equation reads dy dt = −t y − t . The solutions are  y = n≥−1 cn tn/3 in C({t1/3 }), depending on c−1 and c3−1 = −1. An algebraic solution y has to be ramified at z = ∞ of order 3 (and thus y is not rational) and is ramified at some more points with ramification of order 2. However we have no idea what the other ramification points for y could be and what the degree of y over C(z) could be. 3. Testing strict equivalence Given are two first order differential equations f1 , f2 over a field K with [K : C(z)] < ∞. Let (X1 , D1 ), (X2 , D2 ) denote the corresponding pairs of a curve over K and a derivation of its function field extending the derivation of K. We present a procedure testing strict equivalence of f1 and f2 . This amounts to testing whether there exists a finite extension L ⊃ K and an isomorphism L × (X1 , D1 ) → L × (X2 , D2 ). As can be seen from the description below, the procedure can be made into an algorithm except for the case that the curves Xj have genus 1 and moreover the derivations Dj have no poles (equivalently, f1 and f2 have genus 1 and satisfy PP). The first step is to compute the genus of X1 and X2 and to apply known algorithms testing whether K × X1 is isomorphic to K × X2 . In case this question has a positive answer we may suppose (after extending K) that X1 and X2 are equal to a curve X over K. The question is now whether there exists an automorphism A of L × X (with [L : K] < ∞) such that AD1 A−1 = D2 . Suppose that the genus of X is ≥ 2. Then the automorphism group of X is finite. Moreover, if K sufficiently large, then L × X has the same group of automorphisms as X for any [L : K] < ∞. There are algorithms computing the automorphism group of X, see the Appendix. Using these one can test whether AD1 A−1 = D2 . Suppose that X has genus one. Let Sj , for j = 1, 2, denote the set of poles of Dj , i.e., Sj consists of the closed points x ∈ X such that Dj (OX,X ) ⊂ OX,x . We may suppose that S1 , S2 ⊂ X(K). An automorphism A with AD1 A−1 = D2 has the property A(S1 ) = S2 . In particular we have to require that #S1 = #S2 . If S1 = ∅, then there are only finitely many possibilities for A (and these are computable). We are left with the case S1 = S2 = ∅. By Theorem 2.3, K(X) = K(x, y), y 2 = x3 + ax + b, a, b ∈ C, ∂ ∂ + hj · y ∂x with hj ∈ K for j = 1, 2. The group of the automorphism of the Dj = ∂z elliptic curve K ×X is a semi-direct product of the normal subgroup X(K) of translations and the finite cyclic group Aut(X, 0), i.e., the automorphism of X which preserves the ∂ ∂ A−1 = ∂z and neutral element 0 ∈ X. The action of A ∈ Aut(X, 0) has the property A ∂z ∂ ∂ −1 2 3 A(y ∂x )A = ζy ∂x with ζ = 1 and for special elliptic curves X also ζ = 1 or ζ 4 = 1. Thus the group Aut(X, 0) poses no problems for the algorithm that we want to produce. ∂ ∂ A−1 = ∂z + Let A ∈ X(K) be translation over a point (x0 , y0 ) ∈ X(K). Then A ∂z x0 ∂ y0 y ∂x

∂ ∂ and Ay ∂x A−1 = y ∂x . Here the second formula is obvious, the first one requires

7

a computation. Thus the problem of strict equivalence amounts to testing whether for a x given h ∈ K the equation y00 = h has a solution (x0 , y0 ) ∈ X(K). 0 Define the map X(K) → ΩK/C by (x0 , y0 ) → dx y0 . This map is additive because of the above formulas, and its kernel is X(C). We start investigating the above problem by assuming that h ∈ K, h = 0 satisfies 0 hdz = dx y0 for some (x0 , y0 ) ∈ X(K). Write K = C(z, h) and L = C(z, x0 , y0 ). Then 0 K ⊂ L ⊂ K. For every σ ∈ Gal(K/K) one has hdz = dσx σy0 . Thus (σx0 , σy0 ) = (x0 , y0 )⊕ (α, β), where ⊕ denotes the addition on the elliptic curve (X, 0) and (α, β) ∈ X(C). It follows that σx0 , σy0 ∈ L. Therefore L ⊃ K is a Galois extension with Galois group G ⊂ X(C). Now (x∗ , y∗ ) := ⊕σ∈G (σx0 , σy0 ) (again ⊕ denotes the addition on the ∗ elliptic curve) has the properties (x∗ , y∗ ) ∈ X(K) and dx y∗ = (#G) · hdz. The group X(K)/X(C) is a Lang–N´eron group (or a relative Mordell–Weil group) of the elliptic curve X. It is known that this group is finitely generated (and free in this special case). Given an algorithm for finding a basis (x1 , y1 ), . . . , (xm , ym ) of Q ⊗Z dxm 1 X(K)/X(C), testing and solving the equation hdz ∈ Q dx y1 + · · · + Q ym is easy. Suppose that X has genus zero. Sj denotes, for j = 1, 2, the set of poles of Dj . We may assume that #S1 = #S2 and S1 , S2 ⊂ X(K). If #S1 ≥ 3, then there are only finitely many automorphisms A with A(S1 ) = S2 and we can test AD1 A−1 = D2 . that S1 = S2 = {0, ∞}, K(X) = Suppose #S1 = #S  2 = n2. Then wecan suppose an u , D(u) = bn un . The only automorphism A of K × X K(u) and D1 (u) = ∗ that preserve {0, ∞} are A(u) = hu±1 with h ∈ K . Write v = hu, then D1 (v) =   h an h1−n v n and it is easy to test whether this equals bn v n for a suitable h. hv + −1 The same holds for the transformation u → hu . Suppose that #S1 = #S2 = 1. Now we may assume S1 = S2 = {∞} and D1 (u), D2 (u) ∈ K[u] have degrees > 2. The only transformations that we have to consider are A(u) = au + b with a, b ∈ K, a = 0. It is easy to test whether for suitable a, b the equality AD1 A−1 = D2 holds. Suppose that S1 = S2 = ∅. For a pair (X, D) we define a closed point x ∈ X to be a zero of D if D maps the maximal ideal of OX,x to itself. For the case K(X) = K(u), D(u) = a0 + a1 u + a2 u2 one has: 0 is a zero if and only if a0 = 0, ∞ is a zero if and only if a2 = 0 ∗ and the point u = b ∈ K , is a zero if and only if b = a0 + a1 b + a2 b2 . Thus the zero’s in K of D are the algebraic solutions of the Riccati equation u = a0 + a1 u + a2 u2 . Using known algorithms for testing and computing algebraic solutions of the Riccati equation (see Appendix) one obtains a classification of the strict equivalence classes of the pairs (K(u), D) with D(u) = a0 + a1 u + a2 u2 , namely: ˜ with D(u) ˜ (a) If D has ≥ 3 zero’s, then D is strict equivalent to D = 0 and moreover all solutions of the Riccati equation are algebraic. (b) If D has two zero’s, then we may suppose that the zero’s are 0, ∞ and thus D(u) = a1 u  ∗ and a1 = bb has no solution in K . (c) If D has one zero, then we may suppose that this zero is ∞ and D(u) = a0 + a1 u and the equation b = a0 + a1 b has no solution in K. (d) If D has no zero’s, then the after changing u we obtain D(u) = a0 −u2 and b +b2 = a0 has no solution in K. 8

Given are two derivations D1 , D2 of K(u) without poles. For the question of strict equivalence we may suppose that they have the same number of zero’s. We can skip case (a). For case (b) one has D1 (u) = a1 u, D2 (u) = a2 u and the only automorphisms A of K(u) that we have to consider are A(u) = h · u±1 . This leads to the equations ∗ h h h = a2 − a1 or h = a1 + a2 for h ∈ K . There are algorithms testing and solving these equations (see the discussion of Question 5 in the Appendix). For case (c), D1 (u) = a0 + a1 u, D2 (u) = b0 + b1 u and the only automorphism to consider is A(u) = au + b with   a, b ∈ K, a = 0. For v = au + b one has D1 (v) = −(a1 + aa )b + b + aa0 + ( aa + a1 )v.  First we have to solve (if possible) aa + a1 = b1 . If we obtain a solution a, then we may suppose a1 = b1 and we have to consider v = b + cv with c ∈ C∗ , b ∈ K such that D1 (v) = ca0 + b − a1 b + a1 v = b0 + a1 v. We are left with an inhomogeneous equation b − a1 b = −ca0 + b0 which is called a Risch equation. We now present an algorithm finding all algebraic solutions for such an equation. Note that an extensive literature on the Risch equation exists, e.g. [Si], [Ra]. However we could not find an answer there for our specific question. Let K = C(z, a1 , a0 , b0 ) and suppose that the equation b − a1 b = −ca0 + b0 has a solution b in a Galois extension of L of K. For an element σ in the Galois group of L/K, σ(b) is a solution of the same equation and thus (σ(b) − b) = a1 (σ(b) − b). Suppose that ∗ the equation f  = a1 f has no solution in K . Then b ∈ K. Let Z denote the curve over C with function field K. For the (possible) solution b one can compute for each point of Z a bound for the pole of b at that point. This produces a positive divisor D and div(b) ≥ −D. space {f ∈ K| div(f ) ≥ −D}. Then one has to Let b1 , . . . , br be a basis of the  vector verify whether the equation cj (bj − a1 bj ) = ca0 + b0 with c1 , . . . , cd ∈ C, c ∈ C∗ has a solution and compute the solution if it exists.  ∗ Suppose that a1 = ff for some f ∈ K . Then one can change a1 to 0 and the equation becomes b = ca˜0 + b˜0 with c ∈ C∗ . Now a possible algebraic solution b lies in the field K = C(z, a ˜0 , ˜b0 ). Again one computes a positive divisor D on the curve of the field K such that a possible solution b satisfies div(b) ≥ −D. As above one can decide whether a solution exists and compute b in case b exists. Finally, we treat the complicated case (d): D1 (u) + u2 = a0 , D2 (u) + u2 = b0 . A direct computation relating a0 and b0 if D1 is strictly equivalent to D2 seems possible. We follow a more sophisticated approach. Consider the equations Y1 = a0Y1 and Y2 = b0 Y2 over the differential field K. If the equations are equivalent then αγ βδ ∈ GL2 (K) exists Y

Y

γ+δu1 with u1 := Y11 . such that Y2 = αY1 + βY1 and Y2 = γY2 + δY2 . Then u2 := Y22 = α+βu 1 Conversely, if u1 + u21 = a0 and u2 + u22 = b0 are equivalent then the same holds for the linear equations. Let P V ⊃ K be its Picard–Vessiot field and let G ⊂ SL2 be its differential Galois group. Since G is connected and the Riccati equation u + u2 = a0 has no algebraic solutions we have G = SL2 . Let L ⊂ P V be the field of invariants under the subgroup {±1} of G. Then L is the Picard–Vessiot field of the second symmetric power M1 of the equation Y  = a0 Y and the corresponding differential Galois group is PSL2 . Let  Sol ⊂ P V denote the solution space of Y  = a0 Y and let T := { yy | y ∈ Sol, y = 0}. Then T is invariant under G and L is the smallest differential subfield of P V containing

9

T , since the normal subgroups of PSL2 are trivial. We note that T is the solution space of the Riccati equation u + u2 = a0 . Assume that D1 and D2 are strictly equivalent. Consider the equation Y  = b0 Y over the differential field K, which also has differential Galois group SL2 . The same equation over the differential field L has infinitely many solutions for the Riccati equation, since D1 and D2 are strictly equivalent. It follows that L is also the Picard–Vessiot field of the second symmetric power M2 of Y  = b0 Y . By a Tannakian argument, the existence of only one irreducible representation of PSL2 of dimension 3 implies that the differential modules K ⊗ M1 and K ⊗ M2 are isomorphic. Let N1 , N2 denote the two dimensional differential modules corresponding to Y  = a0 Y and Y  = b0 Y . Using ideas from [vH-vdP], Proposition 3.1, applied to the case of the differential field K, one can show the existence of an isomorphism K ⊗ N1 → K ⊗ N2 . For completeness we give details. Lemma 3.1. Let A1 and A2 be two-dimensional differential modules over K with differential Galois group SL2 . Suppose that the second symmetric powers Bj = sym2 Aj , j = 1, 2 are isomorphic. Then A1 and A2 are isomorphic. Proof. Choose a basis v1 , v2 of A1 such that ∂(v1 ∧ v2 ) = 0. Then B1 has basis w1 = v12 , w2 = v22 , w3 = v1 v2 . Put F1 = w32 − w1 w2 ∈ sym2 (B1 ). Then ∂F1 = 0, KF1 is the only 1-dimensional submodule of sym2 B1 and F1 is unique up to multiplication by an element in C∗ . Further KF1 is the kernel of the canonical surjective morphism of differential modules sym2 B1 → sym4 A1 . Let F2 ∈ sym2 B2 satisfy ∂F2 = 0 and F2 = 0. Let φ : B1 → B2 be an isomorphism of differential modules. Then φ induces an isomorphism sym2 (φ) : sym2 B1 → sym2 B2 which sends F1 to cF2 for some c ∈ C∗ . Indeed, F1 and F2 are unique up to multiplication by a scalar. Since K is algebraically closed there exists a K-linear bijection ψ : A1 → A2 with sym2 (ψ) = cφ for some c ∈ C∗ . From φ◦∂ = ∂ ◦φ it easily follows that ψ ◦∂ = ∂ ◦ψ. In other words ψ : A1 → A2 is an isomorphism of differential modules. Testing whether K ⊗ N1 and K ⊗ N2 are isomorphic amounts to finding a non trivial algebraic solution of the 4-dimensional differential module M := Hom(N1 , N2 ) over K. An explicit algorithm for this is the following. Suppose that an isomorphism f : K ⊗ N1 → K ⊗ N2 exists. Then f is unique up multiplication by an element in C∗ . Further, Kf ⊂ K ⊗ M is a 1-dimensional submodule, invariant under Gal(K/K). Hence M has a 1-dimensional submodule N and this 1-dimensional submodule is unique. Now M is semi-simple, because N1 and N2 ˜ . The projection P : M → N ⊂ M with kernel M ˜ is are simple. Then M = N ⊕ M an element of the C-algebra ker(∂, End(M )) and can be computed. If the 1-dimensional N ⊂ M is found, then one writes N = Ke with ∂(e) = he and h ∈ K. Finally one has  ∗ to solve (if possible) aa = h with a ∈ K . 4. Strict equivalence with an autonomous equation Given is an equation f over K, its differential algebra K[s, t, d1 ] and the pair (X, D). We present a procedure which investigates whether (X, D) is strictly equivalent to an autonomous equation, i.e., a pair (X0 , D0 ) of a curve over C and a derivation D0 of C(X0 ). This is a complete algorithm in many cases; however, in one special case (namely, a curve 10

X of genus one with a derivation D without poles) the problem is reduced to being able to answer Question 5 discussed in the Appendix. The first step is to test whether K ×K X is isomorphic to some K ×C X0 . Algorithms for this question are known, see Appendix, question 2. Now we assume that X = K ×C X0 and we have to investigate whether there exists an automorpism A of L ×C X0 (where L is a finite extension of K) such that the derivation ADA−1 leaves the subfield C(X0 ) of L(X0 ) invariant. Suppose that the genus of f is ≥ 2. Then X0 and L ×C X0 have the same finite group of automorphisms. In particular, any automorphism A leaves C(X0 ) invariant and we have just to test whether D leaves C(X0 ) invariant. Suppose that f has genus one. Consider X0 as an elliptic curve with neutral element 0. Let S be the set of poles of D, i.e., the closed points x ∈ X such that D(OX,x ) ⊂ OX,x . After extending K we may assume S ⊂ X0 (K). We note that for an automorphism A one has A−1 (OX,x ) = OX,Ax . Thus the set of poles of ADA−1 is A(S). Therefore the first step to do is to produce an automorphism A of X such that A(S) ⊂ X0 (C). Suppose that S has at least one point x. We may suppose that x = 0 ∈ X0 (C). Further we have only to consider derivations ADA−1 with A(0) = 0. Since C(X0 ) is invariant under A it suffices to test whether C(X0 ) is invariant under D. Suppose that S = ∅. Then K(X) = K(x, y) with y 2 = x3 + ax + b, a, b ∈ C and and ∂ ∂ + h · y ∂x with h ∈ K. In order to test strict equivalence with an autonomous D = ∂z x

equation one needs, according to §3, to solve an equation h + c = y00 with c ∈ C and (x0 , y0 ) ∈ X(K). Let K = C(z, h). A necessary condition is that (h + c)dz is a holomorphic differential on the curve Z with function field K. This determines c. As in §3, we ∗ have to solve N (h + c)dz = dx y∗ for some element (x∗ , y∗ ) ∈ X(K) and an integer N ≥ 1. If an explicit basis of the Lang–N´eron group Q ⊗Z X(K)/X(C) can be computed, then ∗ one can test and solve the equation N (h + c)dz = dx y∗ . Suppose that the genus of f is zero. The function field of X0 is C(u) and D(u) is some element of K(u). Let S denote again the set of poles of D. We may suppose that S ⊂ X0 (K). Suppose that #S ≥ 3. After an automorphism of X we may suppose that S contains 0, 1, ∞. For strictly equivalence with an autonomous equation we just have tot test whether D(u) ∈ C(u). that #S = 2. Then we may suppose that S = {0, ∞}. Thus D(u) =  Suppose n a u ∈ K[u, u−1 ] and we have to consider only the automorphisms A with A(0) = n n  ∗ 0, = ∞. Thus Au = hu for some h ∈ K . Write v = hu. Then D(v) = hh v + A(∞) −n+1 n −1 v . We have to test whether for a suitable h one has D(v) ∈ C[v, v ]. Let n0 an h be minimal with an = 0 and n1 maximal with an = 0. Then n0 < 0, n1 > 2. One obtains the conditions an0 h−n0 +1 , an1 h−n1 +1 ∈ C∗ . Substitution of, say, h = c · (an0 )1/(n0 −1) with c ∈ C∗ leads to a solution of the above question.  Suppose that #S = 1. Then we may suppose that S = {∞} and D(u) = n≥0 an un . We have to search for  v = au + b with a, b ∈ K, a = 0, such that D(v) ∈ C[v]. Now n an ( v−b D(v) = b + a v−b a +a a ) . We note that the maximal n1 with an = 0 satisfies 11

1/(n −1)

n1 > 2. Then a = c · an 1 for some c ∈ C∗ . The coefficient of v n1 −1 in the expression for D(v) yields a formula for b. Substitution leads to a solution of the question. Suppose that S = ∅. We have D(u) = a0 + a1 u + a2 u2 . An autonomous E can be normalized to E(v) = λv with λ ∈ C. A necessary condition is that D has at least two zero’s. Thus one has to compute algebraic solutions of u = a0 + a1 u + a2 u2 . Suppose that we found two of these, then D is strictly equivalent to a D of the form D(u) = a1 u.  ∗ Finally we have to solve the equation a1 = λ + bb for some b ∈ K and λ ∈ C. This amounts to producing λ ∈ C such that (a1 − λ)dz has at most poles of order one. This determines λ. If this meets success, then we have to apply a known algorithm (see [Ri] ∗ and [B-D]) to test and solve the equation db b = (a1 − λ)dz with b ∈ K . 5. Autonomous equations As before, we associate to an irreducible autonomous equation f (y  , y) = 0 (it is assume that both y and y  are present in f ) the pair (X, D) where the complete, irreducible, smooth curve X over C has function field C(y1 , y0 ) with equation f (y1 , y0 ) = 0 and D is the meromorphic vector field on X determined by D(y0 ) = y1 . Two pairs (X1 , D1 ), (X2 , D2 ) are called isomorphic if there exists an isomorphism φ : X1 → X2 of complex curves such that φ∗ D1 = D2 . Theorem 5.1. The map f → (X, D) yields a bijection between strict equivalence classes of autonomous equations and isomorphism classes of pairs (X, D), where X is a curve over C and D a non zero meromorphic vector field. We note that the question whether two pairs (X1 , D1 ), (X2 , D2 ) are isomorphic has an algorithmic answer. Indeed, according to the Appendix there is an algorithm testing whether X1 and X2 are isomorphic. If they are, the algorithm finds an explicit isomorphism. This reduces the problem to the case X = X1 = X2 . Here we need to test whether φ∗ D1 = D2 for a suitable isomorphism φ. If the genus of X is at least two, the finitely many automorphisms of the curve are known explicitly. For smaller genus, the form of φ is known and the test is equivalent to determining whether a finite system of polynomial equations over the complex numbers has a solution. The proof of Theorem 5.1 will be given in 5.2 and 5.4. Lemma 5.2. Every pair (X, D), as in Theorem 5.1, is associated to some autonomous equation f (y  , y) = 0. Proof. Let g ∈ C(X) satisfy D(g) = 0. Choose a closed point x ∈ X such that g has no pole at x, ordx (g − g(x)) = 1 and ordx (D(g)) = 0. Let π denote a local parameter X,x = C[[π]] and ordx (D(π)) = 0. Let p be a prime number such that at x. Then O p > 2 · genus(X) + 2 and let f ∈ C(X) have a pole of order p at x and no further poles. Then [C(X) : C(f )] = p. If D(f ) ∈ C(f ), then C(f, D(f )) = C(X) since p is a prime number. ) ∈ C(f ) ⊂ C(( f1 )) = C((π p )) ⊂ C((π)). This Suppose that D(f ) ∈ C(f ). Then D(f f ) contradicts the fact that ordx ( D(f f ) = −1.

Remark 5.3. An irreducible order one equation f (y  , y, z) = 0 over a finite field extension K of C(z) induces a pair (X, D) of a curve X over K and a derivation D of the 12

function field of X/K. The proof of Lemma 5.2 extends to this non autonomous case. ˜ of The statement is: For a given pair (X, D) over K, there exists a finite extension K  ˜ ˜ K and an irreducible order one equation f (y , y, z) = 0 over K which induces K ×K X equipped with the unique extension of D.  Let (X, D) be a pair as in Theorem 5.1 and let K be a finite extension of C(z). We denote by K × (X, D) the curve K ×C X with function field K ⊗C C(X) equipped with d on K and D+ = D on C(X). We note that D+ the derivation D+ defined by D+ = dz is not a meromorphic vector field since it is not zero on K. Lemma 5.4. Let φ : K × (X1 , D1 ) → K × (X2 , D2 ) be an isomorphism. Then there exists an isomorphism (X1 , D1 ) → (X2 , D2 ). Proof. The isomorphism φ : K × X1 → K × X2 induces an isomorphism φ1 : spec(R) × X1 → spec(R) × X2 for some finitely generated C-algebra R with field of fractions K. After dividing by a maximal ideal of R we find an isomorphism X1 → X2 . In the sequel we identify X1 and X2 with some X. It is given that some automorphism φ of K × X has the property D2+ = φD1+ φ−1 . We have to show that there exists an automorphism ψ of X with D2 = ψD1 ψ −1 . If the genus of X is ≥ 2, then K × X and X have the same finite group of automorphisms and there is nothing to prove. Let the genus of X be one. After choosing a point 0 ∈ X we regard X as the elliptic d with curve with equation y 2 = x3 + ax + b. Further, for j = 1, 2, one has Dj = hj · y dx + −1 h1 , h2 ∈ C(X). Suppose that K × X has an automorphism A such that AD1 A = D2+ , ∂ ∂ where Dj+ = ∂z + hj · y ∂x for j = 1, 2. We may suppose that A is a translation on K × X ∂ over a point (x0 , y0 ) ∈ X(K). Now A ∂z A−1 = x0

∂ ∂z

+

x0 y0

∂ ∂ · y ∂x and A(h1 · y ∂x )A−1 =

∂ . Hence A(h1 ) + y0 = h2 . A(h1 ) · y ∂x Suppose that h1 has a pole q ∈ X(C), then A(q) is a pole of h2 . Thus A(q) ∈ X(C) and A is a translation over a point in X(C). x Suppose that h1 has no poles, then the same holds for h2 and y00 = h2 − h1 = c ∈ C. Suppose c = 0. Then (x0 )2 = c2 y02 = c2 (x30 + ax0 + b). The non constant solutions of this Weierstrass equation are transcendental (since they are doubly periodic), contradicting the algebraicity of x0 .

Suppose that X has genus zero. Write C(X) = C(y) and Dj (y) = hj ∈ C(y) and ∂ ∂ + hj ∂y for j = 1, 2. We note that the poles of Dj+ on K × X coincide with the Dj+ = ∂z poles of Dj on X. Suppose that A is an automorphism of K ×X with AD1+ A−1 = D2+ . Then A maps the poles of D1 to the poles of D2 . If D1 has at least three poles, then A is an automorphism of X. If D1 has two poles, then the same holds for D2 and we may suppose that these poles are 0, ∞. Then h1 , h2 ∈ C[y, y −1 ] and A(y) = f y or A(y) = f y −1 with f ∈ K ∗ . We  may restrict to the first case and obtain then the equality − ff y + f −1 h1 (f y) = h2 (y). This implies f  = 0 and f ∈ C∗ . The case where D1 has only one singularity is similar. 13

Finally, suppose that D1 has no poles. Then C(y) = C(v) holds with D1 v = λ1 v and λ1 ∈ C∗ . Thus we may assume Dj (y) = λj y with λ1 , λ2 ∈ C∗ . We note that the only zero’s of Dj+ on K × X are 0 and ∞. Hence the automorphism A has the form A(y) = f y or A(y) = f y −1 with f ∈ K ∗ . We may restrict ourselves to the case A(y) = f y. Then one  obtains the equality λ1 − λ2 = ff . This equation has no algebraic solution if λ1 = λ2 . 6. Appendix Here we briefly discuss algorithmic aspects of the questions of §1. Algorithms for Question Q1, i.e., testing whether two curves X1 , X2 of genus g ≥ 2 over an algebraically closed field are isomorphic, are known. Moreover there are algorithms computing the automorphism group of a curve of genus ≥ 2 over an algebraically closed field. A description of such a procedure is given in [H2, Algorithm 4]; this has been implemented in Magma, see also [Ma]. For hyperelliptic curves X one can investigate the canonical morphism X → P1 and for other curves the canonical embedding. For any curve of genus ≥ 2 one can use its tricanonical embedding and also its set of Weierstrass points. The special case of smooth plane curves is descibed in [D]. We refer to [H2, Mu, P] and the related manuscript [LRS]. A different approach to question (1) is provided by the coarse moduli space of curves of a genus g ≥ 2 over an algebraically closed field. Generators of the function field of this moduli space can be computed and these can be used for testing whether two curves of genus g are isomorphic. (see e.g. [R] for g = 3). Question Q2, i.e., deciding whether a curve X, say, over, C(z) descends to C, can be handled by the same type of algorithms. Indeed, let X0 over C be a (smooth) specialization of X. The problem is reduced to testing whether X and X0 are isomorphic over C(z). Question Q3, computing algebraic solutions of the Riccati equation u + u2 = r where r lies in a finite extension K of C(z), has been solved by J. Kovacic for the case of K = C(z) [K]. For a proper finite extension K of C(z), the paper [U-W] provides some information. An algorithm is given in [Si] for computing Liouvillian solutions in a very general situation. However, here we are interested in finding all algebraic solutions. We present here an algorithm for finding all algebraic solutions to u + u2 = r. Put K = C(z, r) and let N/K be the two dimensional differential module corresponding to the equation y  = ry. Solutions of the Riccati equation u + u2 = r correspond to 1-dimensional submodules of K ⊗K N . We consider three cases: (a). Suppose that there are at least three 1-dimensional submodules of K ⊗K N . This is equivalent to N having a finite differential Galois group. The latter can be tested by trying to find factors of the symmetric powers symn N for n = 1, 2, 4, 6, 12, as in the Kovacic algorithm. We may suppose that the module N (and its symmetric powers) are semisimple. Using the “eigenring” ker(∂, End(symn N )) one produces its irreducible submodules. For n = 1 one decides whether the possible 1-dimensional summands have a finite Galois group (i.e., whether the associated equation y  = ay has a non trivial algebraic solution, [Ri], [B-D]). If there are no 1-dimensional summands and if for n = 2 the eigenring is non trivial, then one computes as in (b2) below the quadratic extension L/K such that L ⊗ N decomposes and continues as in the case n = 1. In case the 14

eigenring for n = 2 is trivial, one considers n = 4, 6, 12 to detect direct summands of symn N . (b). Suppose that K ⊗ N has precisely two 1-dimensional submodules. (b1). Suppose that both submodules are defined over K, then one can find these factors by computing the eigenring ker(∂, End(N )). (b2). Suppose that both 1-dimensional submodules are defined over a quadratic extension K2 = K(w) of K with w2 ∈ K. Let K2 ⊗K N = A1 ⊕ A2 with A1 = K2 e1 , ∂e1 = (a + bw)e1 , A2 = K2 e2 , ∂e2 = (a − bw)e2 . Then one can find the 1-dimensional factor Ke1 ⊗ e2 of sym2 N and ∂e1 ⊗ e2 = (2a) · e1 ⊗ e2 . Further Λ2 N = Kwe1 ∧ e2 and s



s

∂(we1 ∧ e2 ) = ( ww + 2a) · (we1 ∧ e2 ) =

f f

s

· (we1 ∧ e2 ) for some f ∈ K ∗ . Then w2 is known −2

2 

(up to multiplication by an element of K ∗ ) by 4a = (ww−2ff 2) . Thus K2 is known and by the “eigenring” method one finds the two factors A1 , A2 of K2 ⊗K N . (c). Suppose that K ⊗K N has only one 1-dimensional submodule. Then this submodule ˜ denotes is defined over K. Let A ⊂ N be this 1-dimensional submodule. Now N N , seen as a differential module of dimension 2d = 2 · [K : C(z)] over C(z). Let A˜ ˜ . By Beke’s algorithm (valid denote the corresponding d-dimensional submodule of N ˜ in for a differential module over C(z)) one can find all d-dimensional submodules of N ˜ of dimension d we test whether r · T = T . parametrized form. For any submodule T ⊂ N If T with this property is found, then T is the 1-dimensional submodule of N that we are looking for. If (a)–(c) yield no results, then there are no 1-dimensional submodules of N . Question Q4 has a well known answer by J. Coates (see [Co]). Question Q5. Unlike the earlier questions, we are not aware of any literature concerning an algorithm computing generators for the Lang–N´eron group E(K)/E(C). The essential case is to treat this problem with C replaced by Q. Moreover, the original problem to decide whether a given differential form hdz has the form N · dx y for some (x, y) ∈ E(K), seems somewhat easier. The special case where K is the function field of an elliptic curve F is already non trivial. In this case one asks for the existence of an isogeny between E and F . We sketch what can be done in the case that the base field is a number field. An effective version of Faltings’ isogeny theorem may provide a key for solving the problem here. Indeed, a delicate reasoning using Faltings’ result determines whether the restriction to a subgroup of finite index of the -adic Galois representations attached to E and F are isomorphic. If they are, then E and F are isogenous, and an explicit isogeny can be found (e.g., by calculating the cyclic subgroups γ of E of increasing order N = 1, 2, 3, . . ., until E/γ and F have equal j-invariant. Then an explicit isogeny E → F of order N is easily determined). The more general situation, namely K = Q(X) for some algebraic curve X, may be approached similarly. Here, the existence of a finite morphism X → E is equivalent to the existence of a non trivial homomorphism J → E where J denotes the Jacobian variety of X. Again, Faltings’ isogeny theorem reduces this problem to determining whether some powers of eigenvalues of Frobenius on E also appear as powers of eigenvalues of Frobenius on J. However, even if the existence of a finite morphism X → E can be settled in this way, we are not aware of any general method to construct such a morphism explicitly. 15

Acknowledgements A preliminary version of this paper was written while the first three authors were guests of Vietnam Institute for Advanced Study in Mathematics (VIASM) in Ha Noi. They thank VIASM for its hospitality. We thank Frans Oort for his interest in this work. We thank the referees for their critical remarks which led to many improvements of the paper. [A-C-F-G] J.M. Aroca, J. Cano, R. Feng, X-S. Gao. Algebraic General Solutions of Algebraic Ordinary Differential Equations. ISSAC ’05. 29-36, ACM, New York, 2005. [B-D] F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math., 101 (1979), 42–76. [Co] J. Coates, Construction of rational functions on a curve. Proc. Cambridge Philos. Soc., 68 (1970) 105–123. [D] Hong Du, On the isomorphisms of smooth algebraic curves, ISSAC ’94 Proceedings of the international symposium on Symbolic and algebraic computation, ACM New York, 1994, 15–19. [H1] F. Hess. An algorithm for computing Weierstrass points, in “ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory”, Lecture Notes in Comput. Sci. 2369, Springer-Verlag, New York, 2002, 357–371. [H2] F. Hess. An algorithm for computing isomorphisms of algebraic function fields, in “ANTS-VI Proceedings of the 6th International Symposium on Algorithmic Number Theory”, Lecture Notes in Comput. Sci. 3076, Springer-Verlag, New York, 2004, 263–271. [vH] M. van Hoeij, Rational Parametrizations of Algebraic Curves using a Canonical Divisor, J. Symb. Comp. 23 (1997), 209–227. [vH-vdP] M. van Hoeij and M. van der Put - Descent for differential modules and skew fields - J. Algebra 296 (2006), no 1, 18-55. [K] J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3–43. [LRS] R. Lercier, C. Ritzenthaler, J. Sijsling, Fast computation of isomorphisms of hyperelliptic curves and explicit descent, preprint, 2012, arXiv:1203.5440. [Ma] Magma computer algebra system. http://magma.maths.usyd.edu.au/magma/overview/2/19/14/ [M] M. Matsuda, First Order Differential Equations. Lecture Notes in Mathematics. Springer Verlag, Berline 1980. [Mu] D. Mumford, Curves and their Jacobians, the University of Michigan Press, Ann Arbor, Mich., 1975. [Mun-vdP] G. Muntingh, M. van der Put. Order one equations with the Painlev´e property. Indag. Math. (N.S.) 18 (2007), 83-95. [P] B. Poonen. Isomorphism problem for commutative algebras and schemes. (April 2010), http://mathoverflow.net/questions/21883/ [vdP-S] M. van der Put, M. F. Singer. Galois theory of linear differential equations. Grundlehren der Math. Wiss. 328, Springer-Verlag, Berlin, 2003. [Ra] C. G. Raab, Definite Integration in Differntial Fields. PhD Thesis, Johannes Kepler Universit¨ at Linz, Austria, 2012. http://www.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf [Ri] R.H. Risch, The solution of the problem of integration in finite terms, Bull. A.M.S. 76 (1970), 605–608. [R] S. van Rijnswou. Testing the Equivalence of Planar Curves. PhD Thesis, Eindhoven University of Technology, 2001 – ISBN 90-386-0861-6. [Si] M.F. Singer, Liouvilian Solutions of Linear Differential Equations with Liouvilian Coeffients, J. of Symbolic Computation, 11(3), (1991), 251-274. [U-W] F. Ulmer, J.-A. Weil, A note on Kovacic’s algorithm. J. Symb. Comput. 22 (1996), 179–200.

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