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Differential equations which come from geometry. (English) Zbl 0536.12017
Groupe Étude Anal. Ultramétrique 10e Année: 1982/83, No. 1, Exp. No. 9, 6 p. (1984).
Let K be an algebraic number field, $$L\in K(x)[D]$$, $$D=d/dx$$, a linear differential operator. We recall the conjecture of Grothendieck: ”If for almost all primes $${\mathfrak p}$$ of K, L has a full set of solutions rational modulo $${\mathfrak p}$$ (in an obvious sense) then L has a full set of algebraic solutions.” If this conjecture holds true for L and if the $${\mathfrak p}$$-adic radius of convergence of the solutions of L at a generic point is 1 for almost all $${\mathfrak p}$$, then L is said to ”come from geometry” (DFG). The author formulates the ’Conjecture $$D_ n':$$ ”Let L be a DFG operator of order n, irreducible over K(x), with a solution w such that w, $$w',...,w^{(n-1)}$$ are algebraically dependent over K(x). Then all the solutions of L are algebraic functions.” The author proves $$D_ 2$$.
Reviewer: F.Baldassarri
##### MSC:
 12H25 $$p$$-adic differential equations 14H05 Algebraic functions and function fields in algebraic geometry 14B12 Local deformation theory, Artin approximation, etc. 14G20 Local ground fields in algebraic geometry
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