On the zeta function of a hypersurface. III
type: Item
instance of: scholarly article
title: On the zeta function of a hypersurface. III
zbMATH Open document ID: 0173.48603
DOI: 10.2307/1970477
published in: Annals of Mathematics. Second Series
publication date: 2017-08-04T00:00:00Z
review text: In Parts I and II [Publ. Math., Inst. Hautes Étud. Sci. 12, 5--68 (1962; Zbl 0173.48601); Ann. Math. (2) 80, 227--299 (1964; Zbl 0173.48602)] the author showed that the zeta-function of a non-singular hypersurface defined over \(\mathrm{GF}(q)\) has the overall form predicted by the Weil conjectures and satisfies the right functional equation. The present paper breaks fresh ground by considering the singular hypersurfaces; except for the rationality, little is known (or even conjectured) about the zeta-function of these. The notations, ideas, and to some extent the methods of the previous papers are assumed as a prerequisite. In the first two papers, finite-dimensional quotient spaces of \(p\)-adic power series were introduced which served as \(p\)-adic cohomology groups for the non-singular hypersurface. (An explicit isomorphism with the deRham cohomology has since been constructed by [\textit{N. M. Katz}, Publ. Math., Inst. Hautes Étud. Sci. 35, 71--106 (1968; Zbl 0159.22502)].) In this paper, analogous cohomology spaces \(H^s(\mathfrak L^\ast)\) are introduced for the singular hypersurface, and the bulk of the paper is devoted to proving they are finite-dimensional. These spaces \(H^s(\mathfrak L^\ast)\) are constructed, as before, from a space \(\mathfrak L^\ast\) of \(p\)-adic Laurent series in several variables, the cohomology being defined via the differential operators \(D_i\) derived from the polynomial \(f\) defining the hypersurface. The proof that the spaces \(H^s(\mathfrak L^\ast)\) are finite-dimensional is by induction on \(s\), the space \(H^s(\mathfrak L^\ast)\) being related to the cohomology space \(H^{s-1}(\mathfrak L_\Gamma{}^\ast)\) of a higher-dimensional hypersurface -- essentially a one-parameter family \(H_\Gamma\) of hypersurfaces, all non-singular except for the given hypersurface \(H_0\). (Thus, the deformation theory given in Part II plays an essential role here, too.) The finite-dimensionality of the space \(H^1(\mathfrak L^\ast)\) com…
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 6756819
Wikidata QID: Q106636682
author: Bernard M. Dwork
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1970477
OpenAlex ID: W4245601853
description: scientific article; zbMATH DE number 6756819