Intégration sur un cycle évanescent. (French) Zbl 0538.13007

Deligne, Pierre

Invent. Math. 76, 129-143 (1984).

For f a formal power series in n indeterminates \(x_ 1,...,x_ n\) over a field \(k\), let \(I(f)\) be the diagonal power series \(\Sigma a_ i,...,i^{t^ i}\). The author proves that if f is an algebraic function of \(x_ 1,...,x_ n\) and that k is of characteristic \(p>0\), then \(I(f)\) is algebraic in t. Results of the same nature had been obtained by H. Furstenberg [J. Algebra 7, 271-277 (1967; Zbl 0175.039)] and J. Denef and L. Lipschitz [”Some remarks on algebraic power series” (preprint)] recently succeeded in extending Furstenberg’s method to cover the case treated here. The aim of the present paper is to offer a geometric explanation of Furstenberg’s results, in terms of what in substance should be p-adic properties of vanishing cycles. In an appendix, signs in the coherent duality formalism are discussed.

MSC:

13F25	Formal power series rings
14C99	Cycles and subschemes
14F30	\(p\)-adic cohomology, crystalline cohomology
13J05	Power series rings

Keywords:

algebraicity of diagonal formal power series; p-adic properties of vanishing cycles

Citations:

Zbl 0175.039

References:

[1]	Furstenberg, H.: Algebraic functions over finite fields. J. of Algebra7, 271-277 (1967) · Zbl 0175.03903 · doi:10.1016/0021-8693(67)90061-0
[2]	Raynaud, M.: Spécialisation du foncteur de Picard. Publ. Math. IHES38, 27-76 (1970) · Zbl 0207.51602
[3]	Raynaud, M.: Anneaux locaux henséliens. Lecture Notes in Mathematics, vol. 169. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0203.05102