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.
For the continuation of this review, see [Zbl 0173.48602].


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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