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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.

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) |