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Index of \(p\)-adic differential operators. III: Application to twisted exponential sums. (English) Zbl 0548.12015

Cohomologie \(p\)-adique, Astérisque 119-120, 191-266 (1984).
The author establishes a formula which permits to compute the index of a differential operator acting on functions analytic in a disk. This is used to compute the dimension of Dwork’s cohomologies associated with differential modules of rank one. The author proves estimates for the \(p\)-adic valuation of the coefficients of the Frobenius matrix and then deduces estimates for the \(p\)-adic valuation of the zeros of the \(L\)-functions associated with exponential sums.
{For part I, see Ann. Math. (2) 101, 280–316 (1975; Zbl 0316.12102), for part II, Duke Math. J. 43, 19–31 (1976; Zbl 0338.12108)}.
[For the entire collection see Zbl 0542.00006.]
Reviewer: Philippe Robba

MSC:

12H25 \(p\)-adic differential equations
14F30 \(p\)-adic cohomology, crystalline cohomology
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols