## 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.

### Citations:

Zbl 0980.53485; Zbl 0834.12005
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