## 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.
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

### Citations:

Zbl 0553.00003; Zbl 0278.14004