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Cohomologie \(p\)-adique pour la fonction \(_3F_2\left(\begin{matrix} a,b_1,b_2 \\ c_1,c_2\end{matrix} ;\lambda\right)\). (French) Zbl 0545.12013

Cohomologie \(p\)-adique, Astérisque 119-120, 51-110 (1984).
[For the entire collection see Zbl 0542.00006. A preliminary version was published in Groupe Étude Anal. Ultramétrique 10e Année 1982/83, No.2, Exp. No.16, 15 p. (1984).]
The function in the title is the classical hypergeometric one. The purpose of this paper is to study this function \(_ 3F_ 2\) and the linear differential equation of order 3, \(_ 3L_ 2\), whose solution is \({}_ 3F_ 2\). The method is the one that B. Dwork used in his book ”Lectures on p-adic differential equations” (1982; Zbl 0502.12021) for studying the hypergeometric function \({}_ 2F_ 1\). The starting point is an integral formula. In fact, two formulae were available: one gives \({}_ 3F_ 2\) by a double integration of an algebraic function and the other gives \({}_ 3F_ 2\) by a simple integration of a function involving \({}_ 2F_ 1\). Here the author uses the second one.
The main results are concerned with: an explicit construction of the differential module associated with \({}_ 3L_ 2\) (an explicit base is given), the symplectic structure (in connection with the dual theory), the p-adic radius of convergence of solutions of \({}_ 3L_ 2\) (but the growth of these solutions is not studied) and strong Frobenius structure for \({}_ 3L_ 2\). Moreover, the results of Dwork’s book that are needed are briefly recalled. Therefore this paper gives a new application of Dwork’s ideas. It was not at all obvious that calculations can be achieved for a ”second level” integral formula. In fact, the calculations are rather intricate.
Reviewer: G.Christol

MSC:

12H25 \(p\)-adic differential equations
34G10 Linear differential equations in abstract spaces
14F30 \(p\)-adic cohomology, crystalline cohomology
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)