## 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)

### Citations:

Zbl 0542.00006; Zbl 0502.12021