Dwork, Bernard On the zeta function of a hypersurface. III. (English) Zbl 0173.48603 Ann. Math. (2) 83, 457-519 (1966). 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 [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 \(H^1(\mathfrak L^\ast)\) comes from results in \(p\)-adic ordinary differential equations, to which a brief chapter is devoted.A second type of cohomology space \(\hat{\mathfrak K}_\infty\subset\mathfrak L^\ast\) is introduced on which the basic endomorphism \(\alpha^\ast\) acts, and which is spanned by the eigenvectors of \(\alpha^\ast\). (In the non-singular case, the characteristic polynomial of \(\alpha^\ast\) is essentially the zeta-function.) Its cohomology is also proved to be finite-dimensional, provided \(f\) has coefficients in an algebraic number field. The representation of \(\alpha^\ast\) on the spaces \(H^s(\hat{\mathfrak K}_\infty)\) leads to a decomposition of \(\det(I-t\alpha)^{\delta^{1+n}}\) into the characteristic polynomials of the endomorphism acting on these spaces.The relation of these cohomology spaces to classical ones is left open, nor are any explicit conjectures about the zeta-function offered, for singular hypersurfaces. In this connection, however, the author remarks that the mapping between the cohomology space and its dual which gave the functional equation for non-singular hypersurfaces still exists in this new theory, and has the same formal relation with \(\alpha^\ast\); however, it is no longer an isomorphism. This as well as other clues to the future direction of the theory are offered at the end of the introduction.For Part IV of this paper, see [Ann. Math. (2) 90, 335–352 (1969; Zbl 0213.47402)]. Reviewer: A. Mattuck (M.R. 35 #194) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 3 ReviewsCited in 11 Documents MSC: 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties Citations:Zbl 0173.48601; Zbl 0173.48602; Zbl 0159.22502; Zbl 0213.47402 × Cite Format Result Cite Review PDF Full Text: DOI