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On the index theorem for \(p\)-adic differential equations. II. (Sur le théorème de l’indice des équations différentielles \(p\)-adiques. II.) (French) Zbl 0929.12003

As in their first work under the same title [Ann. Inst. Fourier 43, 1545-1574 (1993; Zbl 0834.12005)], the authors pursue in this paper Robba’s program on indices of \(p\)-adic differential equations. The remarkable results they obtain here bring the theory of \(p\)-adic differential equations to the level of its complex counterpart in the Fuchsian case. Following P. Robba’s intuition, the \(p\)-adic analogue of the notion of regular singularity is defined as an analytic differential module \(M\) over an annulus \(C_{r,R}=\{z\in\mathbb{C}_p,r<| z|<R\}\), whose generic radii at all \(\rho\in]r,R[\) are maximal.
The authors give a new presentation of the \(p\)-adic exponents of \(M\) (which were introduced in their first paper; see F. Loeser’s notes in Séminaire Bourbaki Vol. 1996/97, Exp. No. 822; Astérisque 245, 57-81 (1997) for yet another presentation, due to Dwork), and prove that if their differences are not Liouville numbers, then \(M\) is, in the same analytic category, a successive extension of rank one differential modules \(z(d/dz)y=\alpha y\), where \(\alpha\) runs through a set of representatives of the exponents of \(M\). Thus, in perfect analogy with the Fuchsian case, the index of \(M\) on the space of analytic functions on \(C_{r,R}\) vanishes.
The paper concludes with a globalization of the above theory to algebraic curves, and with a striking \(p\)-adic analogue of Riemann’s existence theorem over the projective line.

MSC:

12H25 \(p\)-adic differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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