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Index of a $$p$$-adic linear differential operator of first order and $$p$$-adic cohomology. (Indice d’un opérateur différentiel linéaire $$p$$-adique d’ordre 1 et cohomologie $$p$$-adique.) (French) Zbl 0517.12013
Groupe Étude Anal. Ultramétrique 9e Année: 1981/82, No. 3, No. J15, 10 pp. (1983).
Translated from the French summary: Motivated by the recent work of B. Dwork [Am. J. Math. 105, 115–156 (1983; Zbl 0517.12012)], A. Adolphson and S. Sperber [Am. J. Math. 106, 549–591 (1984; Zbl 0552.12010)] on $$p$$-adic cohomologies associated with mixed exponential sums, I will give in this exposé a systematic study of analytic $$p$$-adic cohomology in one variable.
All proofs and a detailed bibliography will appear in a forthcoming paper [“Index of $$p$$-adic differential operators. I–III,” (I) Ann. Math. (2) 101, 280–316 (1975; Zbl 0316.12102), (II) Duke Math. J. 43, 19–31 (1976; Zbl 0338.12108), (III) Cohomologie $$p$$-adique, Astérisque 119–120, 191–266 (1984; Zbl 0548.12015)].

##### MSC:
 12H25 $$p$$-adic differential equations 14F30 $$p$$-adic cohomology, crystalline cohomology
##### Keywords:
analytic $$p$$-adic cohomology in one variable
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