## Un théorème de transfert pour les discques singuliers réguliers.(French)Zbl 0553.12014

Cohomologie p-adique, Astérisque 119-120, 151-168 (1984).
[For the entire collection see Zbl 0542.00006.]
Let us consider a differential system x d/dx(U)$$=G U$$ where the entries of the matrix G are analytic elements (i.e. uniform limits of rational fractions without poles in the disk $$| x| <1$$ of $${\mathbb{C}}_ p)$$. Moreover suppose that the differences of the eigenvalues of G(0) are not Liouville numbers. Consider the formal solution near 0: U$$=Y x^{G(0)}$$. This paper is devoted to show that the matrix Y is analytic in $$0<| x| <1$$ if (and only if) there exists an analytic in the generic disc D(t,1) solution of the system. This ”transfer theorem” was known first for ordinary discs (i.e. $$G(0)=0)$$ from Dwork’s works on p-adic differential equations. An application to an index theorem of Adolphson is also given. The proof is based on Frobenius structures and on the splitting of matrices in singular factors.

### MSC:

 12H25 $$p$$-adic differential equations

Zbl 0542.00006