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**Applications of Padé approximations to the Grothendieck conjecture on linear differential equations.**
*(English)*
Zbl 0565.14010

Number theory, Semin. New York 1983-84, Lect. Notes Math. 1135, 52-100 (1985).

[For the entire collection see Zbl 0553.00003.]

The Grothendieck conjecture on linear differential equations concerns operators of the form \(L=\sum^{n}_{i=0}a_ i(x)(d/dx)^ i\) with \(a_ i(x)\in K(x)\), K an algebraic number field. It asserts that if for almost all primes \({\mathfrak p}\) of K the reduced (mod p) equation \(L_ p\) has a full set of solutions in \(\bar K_ p(x)\) \((\bar K_ p=\) residue field at \({\mathfrak p})\), then L should have a full set of algebraic solutions. A consequence of the previous conjecture is the following statement, formulated as a question by N. Katz [Invent. Math. 18, 1-118 (1972; Zbl 0278.14004)]: (Log) Let C be a complete non singular curve defined over K and \(\omega\) be a differential on C rational over K with only simple poles and residues in \({\mathbb{Q}}\). Suppose that for almost all primes \({\mathfrak p}\) of K the reduced (mod p) differential \(\omega_ p\) is logarithmic on the reduced curve \(C_ p\). Then an integral multiple \(n\omega\), \(n\in {\mathbb{Z}}\setminus \{0\}\), of \(\omega\) is logarithmic on C.

In this beautiful paper the authors prove statement (Log); their methods seem to offer powerful tools to approach the general Grothendieck conjecture. We list now the 3 basic ingredients of the authors’ proof. (1) The assumptions of the Grothendieck conjecture imply strong estimates on the denominators of the coefficients of \(y(x)^ j\), \(j=1,2,...\), if y(x) is a solution of L with algebraic initial values. - (2) The theory of abelian functions allows one to express a solution y of \(dy/y=\omega\) (in Log) in terms of meromorphic functions (of the integrals of the first kind on C) of bounded order of growth. - (3) A striking result on Padé approximations (main theorem 5.2 of the paper) shows that a function y(x) satisfying (1) and (2) must be algebraic.

The authors also give applications to the Lamé equation (with half- integral exponent difference at infinity): this equation satisfies the Grothendieck conjecture, since its essential part is the equation satisfied by a ratio of solutions and this is a first order linear d. e. on an elliptic curve.

The Grothendieck conjecture on linear differential equations concerns operators of the form \(L=\sum^{n}_{i=0}a_ i(x)(d/dx)^ i\) with \(a_ i(x)\in K(x)\), K an algebraic number field. It asserts that if for almost all primes \({\mathfrak p}\) of K the reduced (mod p) equation \(L_ p\) has a full set of solutions in \(\bar K_ p(x)\) \((\bar K_ p=\) residue field at \({\mathfrak p})\), then L should have a full set of algebraic solutions. A consequence of the previous conjecture is the following statement, formulated as a question by N. Katz [Invent. Math. 18, 1-118 (1972; Zbl 0278.14004)]: (Log) Let C be a complete non singular curve defined over K and \(\omega\) be a differential on C rational over K with only simple poles and residues in \({\mathbb{Q}}\). Suppose that for almost all primes \({\mathfrak p}\) of K the reduced (mod p) differential \(\omega_ p\) is logarithmic on the reduced curve \(C_ p\). Then an integral multiple \(n\omega\), \(n\in {\mathbb{Z}}\setminus \{0\}\), of \(\omega\) is logarithmic on C.

In this beautiful paper the authors prove statement (Log); their methods seem to offer powerful tools to approach the general Grothendieck conjecture. We list now the 3 basic ingredients of the authors’ proof. (1) The assumptions of the Grothendieck conjecture imply strong estimates on the denominators of the coefficients of \(y(x)^ j\), \(j=1,2,...\), if y(x) is a solution of L with algebraic initial values. - (2) The theory of abelian functions allows one to express a solution y of \(dy/y=\omega\) (in Log) in terms of meromorphic functions (of the integrals of the first kind on C) of bounded order of growth. - (3) A striking result on Padé approximations (main theorem 5.2 of the paper) shows that a function y(x) satisfying (1) and (2) must be algebraic.

The authors also give applications to the Lamé equation (with half- integral exponent difference at infinity): this equation satisfies the Grothendieck conjecture, since its essential part is the equation satisfied by a ratio of solutions and this is a first order linear d. e. on an elliptic curve.

Reviewer: F.Baldassarri

### MSC:

14H05 | Algebraic functions and function fields in algebraic geometry |

12H25 | \(p\)-adic differential equations |

41A21 | Padé approximation |

14K20 | Analytic theory of abelian varieties; abelian integrals and differentials |

14G20 | Local ground fields in algebraic geometry |