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The paper serves as an introduction to etale cohomology of rigid analytic spaces. A number of basic results are proved, e.g. concerning cohomological dimension, base change, invariance for change of base elds, the homotopy axiom and comparison for etale cohomology of algebraic varieties. The methods are those of classical rigid analytic geometry and along… (More)

1 Statement of the result Consider the second order linear differential equation fl=ry, withrc Q(z). Let Qc~ denote the algebraic closure of the field of rational numbers Q and let G denote thedifferential Galois group over Qcl(z) of this equation. Then G C S/(2, Qct). F~r any solution y # O of the equation the element u = $ satisfies the Riccati equation… (More)

a r t i c l e i n f o a b s t r a c t 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… (More)

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlevé equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices,… (More)

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann– Hilbert correspondence between moduli spaces of rank two connections on P 1 and moduli spaces for the mon-odromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an… (More)