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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 √Čtud. 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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