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On the zeta function of a hypersurface. I. (English) Zbl 0173.48601
The author continues his strikingly original work on the zeta function of an algebraic variety defined over a finite field. Here he considers a nonsingular hypersurface $$H$$ of degree $$d$$ in projective $$n$$-space over the field $$k=GF(q)$$. The Weil conjectures predict that its zeta function has the form $\zeta(H,t)=P(t)^{(-1)^n}/\prod_0^{n-1}(1-q^it), \tag{1}$ where $$P(t)$$ is a polynomial of predicted degree. This paper gives a proof covering all cases except when $$\text{char}(k)=2$$ and $$d$$ is even. The object studied is a certain linear operator $$\alpha$$ acting on a space of $$p$$-adic power series, whose connection with the $$\zeta$$-function is taken over from an earlier paper (cited below), with the addition of (2) below. A partial spectral theory is developed for $$\alpha$$ here, and a key role is also played by a Koszul-type complex associated with a set of differential operators on the space of power series. Some theorems in $$p$$-adic analysis are included.
Let $$Q_p$$ be the rational $$p$$-adic field, and $$\Omega$$ the completion of the algebraic closure of $$Q_p$$. We consider first an operator $$\alpha$$ of the general type needed for the problem. This $$\alpha$$ can be defined on the space of all power series $$\Omega(X_0,\cdots,X_{n+1})$$, but in order that its spectrum not be all of $$\Omega^\ast$$, its action is restricted to a subspace $$L(\kappa)$$ of series whose coefficients grow in ordinal at least at rate $$\kappa$$ (a real number). Fix a suitable series $$F$$ in $$L(\kappa)$$, and define an operator $$\alpha$$ on $$L(q\kappa)$$ by $$\alpha(G)=\Psi\cdot(F\cdot G)$$, where $$\Psi$$ is the $$q$$th root operator on $$\Omega(X)$$, sending monomials of the form $$X^{(qu)}$$ into $$X^{(u)}$$ and the others into 0. A ”characteristic series” $$\chi_F$$ is introduced as the limit of the characteristic polynomials of $$\alpha$$ acting on truncations of $$L(q\kappa)$$. The spectral theory for $$\alpha$$ is now done assuming that the coefficients of $$F$$ lie in a finite extension $$K_0$$ of $$Q_p$$. To each zero $$\lambda^{-1}$$ of $$\chi_F$$ of multiplicity $$s_\lambda$$, it associates the primary subspace $$W_\lambda$$, of $$\dim s_\lambda$$, which is the kernel of $$(I-\lambda^{-1})^s$$ for all $$s\geq s_\lambda$$. If we restrict the coefficient field to $$K_0$$, then if $$\lambda^{-1}$$ is not a zero of $$\chi_F$$, then $$I-\lambda^{-1}\alpha$$ is surjective, with the obvious generalization using $$s_\lambda$$.
This spectral theory was subsequently generalized and simplified by J.-P. Serre [Publ. Math., Inst. Hautes Étud. Sci. 12, 69–85 (1962; Zbl 0104.33601)].
For the application to $$\zeta(H,t)$$, various choices for $$F$$ may be made. If $$\overline f(X)$$ is the defining polynomial for $$H$$ over $$k$$, let $$f(X)$$ be the unique polynomial over $$\Omega$$ reducing $$\text{mod}\,p$$ to $$\overline f$$ and whose coefficients are $$(q-1)$$st roots of unity. Put $$H=\gamma X_0f$$, where $$\gamma^{q-1}=1$$, and then we fix $$F=(\exp H)^\delta$$, where in general $$G(X)^\delta$$ is defined to be $$G(X)/G(X^q)$$. Actually, other $$F$$’s are usable and later one must switch from one to another.
From the author’s earlier paper [Am. J. Math. 82, 631–648 (1960; Zbl 0173.48501)] the basic connection between $$\zeta(H,t)$$ and $$\alpha$$ is (letting $$H'$$ be the hypersurface $$X_1X_2\cdots X_{n+1}=0$$) $\zeta(H-H',qt)=\chi_F{}^{-(-\delta)^{n+1}}(1-t)^{-(-\delta)^n}.$ Let $$A$$ be a nonempty subset of $$S=\{1,\cdots,n+1\}$$ and $$H_A$$ be the hyperplane obtained by intersecting $$H$$ with the hyperplanes $$X_i=0$$, $$i\in A$$. We can assume that $$H_A$$ is nonsingular for all $$A$$. Letting the equation (1) for $$H_A$$ define the rational function $$P_A(t)$$, one can deduce formally that $\chi_F{}^{\delta^{n+1}}=(1-t)\prod_AP_A(qt). \tag{2}$ Says the author: “We believe this equation is quite significant since $$\chi_F$$ is entire even if $$H$$ is singular.”
To show $$P_S(qt)$$ is a polynomial, the essential first step is to show that $$\chi_F{}^{\delta^{n+1}}$$ is a polynomial of degree $$d^n$$. This is true because there is a finite-dimensional quotient space $$\mathfrak W$$ of $$L(q\kappa)$$ on which $$\alpha$$ acts (as $$\overline\alpha$$, say) and $\chi_F{}^{\delta^{n+1}}=\det(I-t\overline\alpha). \tag{3}$ The idea of the proof of this is to introduce differential operators on $$L(q\kappa)\colon D_iG=X_i\,\partial G/\partial X_i+HG\;(i=1,\cdots,n+1)$$. We have easily $$\alpha\circ D_i=qD_i\circ\alpha$$, showing that if $$\lambda^{-1}$$ is an eigenvalue of $$\alpha$$, so is $$q\lambda^{-1}$$, and in fact $$D_i(W_\lambda)\subset W_{\lambda/q}$$; thus $$\alpha$$ acts (as $$\overline\alpha$$) on $$\mathfrak W=L(q\kappa)/\sum D_iL(q\kappa)$$. The natural projection $$L(q\kappa)\rightarrow\mathfrak W$$ carries a primary subspace $$W_\lambda$$ onto the eigenspace $$\mathfrak W_\lambda$$ if $$\lambda^{-1}$$ is also an eigenvalue of $$\overline\alpha$$ (otherwise onto 0), and all eigenspaces of $$\overline\alpha$$ arise this way. Moreover, it induces an isomorphism $$W_\lambda/\sum D_iW_{\lambda/q}\rightarrow\mathfrak W_\lambda$$. All this follows from the spectral theory. Letting $$\dim\mathfrak W_\lambda=b_\lambda$$, what must be proved is therefore the first equality of $\chi_F{}^{\delta^{n+1}}=\prod_\lambda(1-\lambda^{-1}t)^{b_\lambda}=\det(I-t\overline\alpha) \tag{3}$ the product being taken over the spectrum of $$\alpha$$. To do this the author uses a complex which is a modification of the exterior algebra complex (here apparently invented ab ovo) $0\rightarrow F_{n+1}\rightarrow F_n\rightarrow\cdots\rightarrow F_0\rightarrow W_\lambda\rightarrow 0,$ where $$F_r=W_{\lambda/q^r}\otimes\Lambda^rE$$. The differentiations in the complex are the usual ones, employing the $$n+1$$ commuting endomorphisms $$D_i$$. If one knows the sequence is exact, the equality (3) follows trivially; but by the usual formalism of these complexes, exactness follows if one knows that for all $$k$$, $$D_k\beta=\sum D_i\beta_i\Rightarrow\beta=\sum D_i\beta_i{}'$$, where $$\beta_i,\beta_i{}'\in W_{\lambda/q}$$ and $$\beta\in W_\lambda$$. This last statement is the crux of the matter; it is proved first when $$\beta,\beta_i,\beta_i{}'$$ are simply in $$L(q\kappa)$$, using elementary but long computations, then the spectral theory is used to put the elements in the right primary subspaces. It still must be shown that $$P_S(qt)$$ is a polynomial. For this purpose a decomposition $$W=\sum W_A{}^A$$ is given so that $$\overline\alpha$$ induces an $$\overline\alpha_A{}^A$$ on each summand, and $\det(I-t\overline\alpha)=\prod_{A\subset S}\det(I-t\alpha_A{}^A)=\prod P_A(qt), \tag{4}$ where the second equality follows from the first, (2), and (3). Now (4) is still valid if $$S$$ is replaced by any subset $$B\subset S$$; the resulting system of relations shows easily that $$P_S(qt)=\det(I-t\overline\alpha_S{}^S)$$, which completes the proof, the degree being calculated via the Koszul resolution. The $$W_A{}^A$$ are obtained as natural quotients (via the $$D_i$$) of spaces $$L_A{}^A(q\kappa)$$ obtained by taking the power series in $$L(q\kappa)$$, setting $$X_i=0$$ for all $$i\not\in A$$, and then taking just the series divisible by $$X_i$$, for all $$i\in A$$. This last condition causes technical complications in the rather involved (in algebra and convergence) calculations in $$L(q\kappa)$$ which establish both the decomposition of $$W$$ and (4) above. The case $$p|d$$ is particularly troublesome and is excluded entirely when $$p=2$$.
Subsequent work by the author has established the missing case ($$p$$ and $$d$$ mentioned above, and proved the conjectured functional equation for $$\zeta(H,t)$$. The location of the zeros of $$P(t)$$ remains open.

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
Keywords:
algebraic geometry
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References:
 [1] B. Dwork, On the rationality of the zeta function of an algebraic variety,Amer. J. Math., vol. 82 (1960), pp. 631–648. · Zbl 0173.48501 [2] J.-P. Serre,Rationalité des fonctions zêta des variétés algébriques, Séminaire Bourbaki, 1959–1960, no 198. [3] A. Weil, Numbers of solutions of equations in finite fields,Bull. Amer. Math. Soc., vol. 55 (1949), pp. 497–508. · Zbl 0032.39402 [4] E. Artin,Algebraic numbers and algebraic functions, Princeton University, New York University, 1950–1951 (Mimeographed notes). [5] W. Gröbner,Moderne Algebraische Geometrie, Wien, Springer, 1949. · Zbl 0033.12706 [6] B. Dwork, On the congruence properties of thezeta function of algebraic varieties,J. Reine angew. Math., vol. 23 (1960), pp. 130–142. · Zbl 0119.36804 [7] S. Lang, Introduction to algebraic geometry,Interscience Tracts, no 5, New York, 1958. · Zbl 0095.15301
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