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On the zeta function of a hypersurface. II. (English) Zbl 1367.14006
The author here continues his work on the Weil conjectures for the zeta function of a non-singular projective hypersurface $$H$$ defined over $$\text{GF}(q)$$. In Part I of this paper For part I, [B. Dwork, Publ. Math., Inst. Hautes Étud. Sci. 12, 5–68 (1962; Zbl 0173.48601)] (cited below as (I), and a prerequisite for this paper), he identified the zeta function $$P(t)$$ of $$H$$ as the characteristic polynomial of a certain endomorphism $$\overline\alpha$$ acting on a space $$\mathfrak W^S$$ of $$q$$-adic power series. In this paper, he proves that $$P(t)$$ satisfies the right functional equation. (The location of its zeros is the deep remaining question.)
The functional equation is a duality statement, asserting that $$P(t)$$ is essentially unchanged by the substitution $$t\rightarrow q^{n-1}/t$$, where $$n-1=\text{dimension}\,H$$. The idea of the paper is to prove this by getting a dual theory to the one given in (I). A space $$\mathfrak K$$ of $$q$$-adic Laurent series (i.e., negative exponents, and growth conditions) is defined; a quotient space $$\mathfrak K/\mathfrak K^S$$ is naturally dual to $$\mathfrak W^S$$, an endomorphism $$\alpha^\ast$$ dual to $$\alpha$$ is naturally defined on $$\mathfrak K/\mathfrak K^S$$, and one gets (1) $$P(t)=\det(1-t\alpha^\ast)$$. The main point is now to give a certain isomorphism (2) $$\overline\theta\colon\mathfrak K/\mathfrak K^S\rightarrow\mathfrak W^S$$ satisfying the condition (3) $$\overline\theta\circ\alpha^\ast\circ\overline\theta^{-1}=q^{n+1}(\overline\alpha)^{-1}$$. The functional equation then follows immediately from (1), (2), and (3).
The definition of a $$\overline\theta$$ satisfying (3) is done in several steps. First one assumes the hypersurface $$H$$ (or rather, a lifting of it to the $$q$$-adics) is given by a diagonalized equation (4) $$f(X)=\sum a_iX_i{}^d=0$$, $$i=1,\cdots,n+1$$, where the definition of $$\overline\theta$$ and proof of (3) use the partial differentiation operators $$D_i$$ of (I) and their duals $$D_i{}^\ast$$. Here everything is very explicit.
For a general hypersurface, one gives $$\overline\theta$$ by viewing $$H$$ as a deformation of (4), writing its equation in the form (5) $$f(X)=a_iX_i{}^d+\Gamma h(X)$$, where $$\Gamma$$ is a $$q$$-adic variable. Thus, (4) corresponds to $$\Gamma=0$$. For each $$\Gamma$$, one gets the corresponding spaces $$\mathfrak K_\Gamma/\mathfrak K_\Gamma{}^S$$ and $$\mathfrak W_\Gamma{}^S$$.
Now if $$\Gamma$$ is near 0 $$q$$-adically, then by using the $$q$$-adic exponential function, the author defines an isomorphism (6) $$T_\Gamma\colon\mathfrak K_0/\mathfrak K_0{}^S\rightarrow\mathfrak K_\Gamma/\mathfrak K_\Gamma{}^S$$ which enables him to define $$\overline\theta_\Gamma$$ as the composite map: (7) $$\overline\theta_\Gamma\colon\mathfrak K_\Gamma/\mathfrak K_\Gamma{}^S\rightarrow\mathfrak K_0/\mathfrak K_0{}^S\underset\theta{_0}\rightarrow\mathfrak W_0{}^S\rightarrow\mathfrak W_\Gamma{}^S$$, the last map being the dual to $$T_\Gamma$$. The proof of (3) is easy now.
Finally, for arbitrary values of $$\Gamma$$, he takes bases for the two spaces $$\mathfrak K_\Gamma/\mathfrak K_\Gamma{}^S$$ and $$\mathfrak W_\Gamma{}^S$$ and shows that the matrix giving $$\overline\theta$$ has entries which are rational functions of $$\Gamma$$. This enables him to extend the definition of $$\overline\theta$$ to all but a finite number of values of $$\Gamma$$. Since $$\overline\theta$$ satisfies (3) (using $$\alpha_\Gamma$$ and $$\alpha_\Gamma{}^\ast$$) for $$\Gamma$$ near 0, it follows by Krasner’s $$p$$-adic analytic continuation theory that it satisfies (3) for all $$\Gamma$$.
Some final remarks. (1) The matrix $$C_\Gamma$$ representing the map (6) satisfies a Picard-Fuchs equation $$\partial C_\Gamma/\partial\Gamma=C_\Gamma B$$. When the hypersurface is a curve, it has been checked for low values of the genus (in this paper, for genus 1 only) that $$C_\Gamma$$ is actually the period matrix for the normalized integrals of the second kind on the curve. (2) A second, less restrictive proof of (3), not using analytic continuation but instead some involved combinatorial arguments, is also given. (3) A final section gives a method for determining the doubtful sign in the functional equation; in particular, if $$n-1$$ is odd, the sign depends only on the degree $$d$$, $$n-1$$, and on $$q$$.
In subsequent work the author considers singular hypersurfaces. Recent unpublished work by Washnitzer, Monsky, and Katz sheds new light on Dwork’s methods.

##### 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
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