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**On the zeta function of a hypersurface. II.**
*(English)*
Zbl 0173.48602

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 [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.

For the continuation of this review, see [Zbl 0173.48603].

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.

For the continuation of this review, see [Zbl 0173.48603].

Reviewer: A. Mattuck (M.R. 32 #5654)