##
**Zeta-functions of varieties over finite fields at \(s=1\).**
*(English)*
Zbl 0567.14015

Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 173-194 (1983).

[For the entire collection see Zbl 0518.00004.]

From the author’s introduction: Let k be a finite field of cardinality \(q=p^ f\). Let \(\bar k\) be a fixed algebraic closure of k. Let X be a smooth projective algebraic variety of dimension d over k such that \(\bar X=X\times \bar k\) is connected. It is by now well-known that the zeta- function of X may be written in the form \(\zeta (X,s)=Z(X,q^{-s})\), where \[ Z(X,T)=\frac{P_ 1(X,T)P_ 3(X,T)...P_{2d-1}(X,T)}{P_ 0(X,T)P_ 2(X,T)...P_{2d}(X,T)} \] where \(P_ i(X,T)=\det (1- \phi_{i,\ell}T)\) is the characteristic polynomial of the endomorphism \(\phi_{i,\ell}\) of the étale cohomology groups \(H^ i(\bar X,{\mathbb{Q}}_{\ell})\) induced by the Frobenius endomorphism \(\phi\) of X. Deligne has shown, that these polynomials \(P_ i(X,T)\) have rational integral coefficients, are independent of \(\ell\), and (the Riemann hypothesis) have complex ”reciprocal roots” of absolute value \(q^{i/2}\). - Motivated by the analogue of the Birch-Swinnerton-Dyer conjecture for Jacobians of curves over function-fields, Tate and Artin produced a conjecture when \(d=2\) relating the behavior of \(P_ 2(X,T)\) as \(T\to q^{-1}\) to various cohomological invariants of X. Moreover, they proved that if the \(\ell\)-part Br(X)(\(\ell)\) of the Brauer group \(H^ 2(X,{\mathbb{G}}_ m)\) of X was finite for one prime \(\ell\) different from p then the \(\ell\)-part of the Brauer group prime to p was finite, and their conjecture was true for X up to a power of p. Milne completed the work of Tate and Artin for surfaces by removing the restrictions concerning the characteristic.

In this paper we propose to begin to extend the work of Tate, Artin, and Milne to varieties of arbitrary dimension. There no longer seems to be any reason to expect a particularly elegant formula for \(P_ 2(X,T)\), so we are led to turn our attention to the full zeta-function Z(X,T). If we let \(\rho_ 1(X)\) be the rank of the Néron-Severi group of X, we now obtain the conjectured formula: \[ Z(X,T)\sim \pm c_ X(1-qT)^{-\rho_ 1(X)}\quad as\quad T\to q^{- 1},\quad where\quad c_ X=\frac{\chi (X,{\mathbb{G}}_ a)}{\chi (X,{\mathbb{G}}_ m)}. \tag{\(*\)} \] This formula, although remarkably simple, needs some explanation. We mean by \(\chi (X,{\mathbb{G}}_ a)\) the Euler characteristic of the sheaf \({\mathbb{G}}_ a\) for the étale (or flat) topology on X, namely \(\chi (x,{\mathbb{G}}_ a)=\#H^ 0(X,G_ a) \#H^ 2(X,{\mathbb{G}}_ a).../(\#H^ 1(X,{\mathbb{G}}_ a) \#H^ 3(X,G_ a)...).\) In view of the standard fact that \(H^ i(X,{\mathbb{G}}_ a)\) is isomorphic to \(H^ i(X,{\mathcal O}_ X)\), where the cohomology groups are now taken with respect to the Zariski topology, we see that \(\chi (X,{\mathbb{G}}_ a)=q^{\chi (X,{\mathcal O}_ X)}\). We would like to define \(\chi (X,{\mathbb{G}}_ m)\) in the analogous way, but run into the obvious problem that the \(H^ i(X,{\mathbb{G}}_ m)\) are not all finite. - However, at least in good cases, it is still possible to define an Euler characteristic. The following seems to be a natural generalization of the usual notion: Suppose that we have a sequence of cohomology groups \(H^ i\), \(i=0,1,2,..\). with the following properties:

(a) The \(H^ i\) are zero for i large.

(b) The \(H^ i\) are finite for all i except \(i=a\), \(i=a+2.\)

(c) \(H^ a\) is a finitely generated abelian group.

(d) \(H^{a+2}\) is the dual of a finitely generated abelian group \(D^{a+2}.\)

(e) There is a natural pairing \(<\;,\;>\) from \(D^{a+2}\times H^ a\to {\mathbb{Q}}.\)

Then we may define the Euler characteristic in the usual way, except that #H\({}^ a\) is replaced by #(H\({}^ a_{tor})\), #H\({}^{a+2}\) is replaced by #(H\({}^{a+2}_{\cot or})=\#(D^{a+2}_{tor})\), and we add a regulator term \(R(G_ m)=\{\det <h_ i,d_ j>\}^{(- 1)^{a+1}}\), where \(\{h_ i\}\) and \(\{d_ j\}\) are bases of \(H^ a\) modulo torsion and \(D^{a+2}\) modulo torsion, respectively. In our case, \(a=1\), and \(H^ 3(X,{\mathbb{G}}_ m)\) ought to be, up to a finite group, the dual of the finitely generated abelian group consisting of 1-cycles modulo numerical equivalence.

In this paper, we show that, (1) (*) is true for X of dimension 0 or 1; (2) (*) is equivalent to Tate’s conjecture for X of dimension 2. In particular (*) is true if the \(\ell\)-part of \(H^ 2(X,{\mathbb{G}}_ m)\) is finite for any single prime \(\ell\), by the results of Tate and Milne; (3) (*) is true up to a power of p for X of any dimension, provided the \(\ell\)-part of \(H^ 2(X,{\mathbb{G}}_ m)\) is finite for any single prime \(\ell \neq p.\)

Along the way we show that \(H^ i(X,{\mathbb{G}}_ m)\) is finite for \(i\neq 1,2\), and 3, and that if in addition \(H^ 2(X,{\mathbb{G}}_ m)(\ell)\) is finite for one prime \(\ell \neq p\), we have \((i) H^ 2(X,{\mathbb{G}}_ m)(non-p)\) is finite, and (ii) \(H^ 3(X,{\mathbb{G}}_ m)\) is the dual of a finitely-generated abelian group (up to p-torsion).

From the author’s introduction: Let k be a finite field of cardinality \(q=p^ f\). Let \(\bar k\) be a fixed algebraic closure of k. Let X be a smooth projective algebraic variety of dimension d over k such that \(\bar X=X\times \bar k\) is connected. It is by now well-known that the zeta- function of X may be written in the form \(\zeta (X,s)=Z(X,q^{-s})\), where \[ Z(X,T)=\frac{P_ 1(X,T)P_ 3(X,T)...P_{2d-1}(X,T)}{P_ 0(X,T)P_ 2(X,T)...P_{2d}(X,T)} \] where \(P_ i(X,T)=\det (1- \phi_{i,\ell}T)\) is the characteristic polynomial of the endomorphism \(\phi_{i,\ell}\) of the étale cohomology groups \(H^ i(\bar X,{\mathbb{Q}}_{\ell})\) induced by the Frobenius endomorphism \(\phi\) of X. Deligne has shown, that these polynomials \(P_ i(X,T)\) have rational integral coefficients, are independent of \(\ell\), and (the Riemann hypothesis) have complex ”reciprocal roots” of absolute value \(q^{i/2}\). - Motivated by the analogue of the Birch-Swinnerton-Dyer conjecture for Jacobians of curves over function-fields, Tate and Artin produced a conjecture when \(d=2\) relating the behavior of \(P_ 2(X,T)\) as \(T\to q^{-1}\) to various cohomological invariants of X. Moreover, they proved that if the \(\ell\)-part Br(X)(\(\ell)\) of the Brauer group \(H^ 2(X,{\mathbb{G}}_ m)\) of X was finite for one prime \(\ell\) different from p then the \(\ell\)-part of the Brauer group prime to p was finite, and their conjecture was true for X up to a power of p. Milne completed the work of Tate and Artin for surfaces by removing the restrictions concerning the characteristic.

In this paper we propose to begin to extend the work of Tate, Artin, and Milne to varieties of arbitrary dimension. There no longer seems to be any reason to expect a particularly elegant formula for \(P_ 2(X,T)\), so we are led to turn our attention to the full zeta-function Z(X,T). If we let \(\rho_ 1(X)\) be the rank of the Néron-Severi group of X, we now obtain the conjectured formula: \[ Z(X,T)\sim \pm c_ X(1-qT)^{-\rho_ 1(X)}\quad as\quad T\to q^{- 1},\quad where\quad c_ X=\frac{\chi (X,{\mathbb{G}}_ a)}{\chi (X,{\mathbb{G}}_ m)}. \tag{\(*\)} \] This formula, although remarkably simple, needs some explanation. We mean by \(\chi (X,{\mathbb{G}}_ a)\) the Euler characteristic of the sheaf \({\mathbb{G}}_ a\) for the étale (or flat) topology on X, namely \(\chi (x,{\mathbb{G}}_ a)=\#H^ 0(X,G_ a) \#H^ 2(X,{\mathbb{G}}_ a).../(\#H^ 1(X,{\mathbb{G}}_ a) \#H^ 3(X,G_ a)...).\) In view of the standard fact that \(H^ i(X,{\mathbb{G}}_ a)\) is isomorphic to \(H^ i(X,{\mathcal O}_ X)\), where the cohomology groups are now taken with respect to the Zariski topology, we see that \(\chi (X,{\mathbb{G}}_ a)=q^{\chi (X,{\mathcal O}_ X)}\). We would like to define \(\chi (X,{\mathbb{G}}_ m)\) in the analogous way, but run into the obvious problem that the \(H^ i(X,{\mathbb{G}}_ m)\) are not all finite. - However, at least in good cases, it is still possible to define an Euler characteristic. The following seems to be a natural generalization of the usual notion: Suppose that we have a sequence of cohomology groups \(H^ i\), \(i=0,1,2,..\). with the following properties:

(a) The \(H^ i\) are zero for i large.

(b) The \(H^ i\) are finite for all i except \(i=a\), \(i=a+2.\)

(c) \(H^ a\) is a finitely generated abelian group.

(d) \(H^{a+2}\) is the dual of a finitely generated abelian group \(D^{a+2}.\)

(e) There is a natural pairing \(<\;,\;>\) from \(D^{a+2}\times H^ a\to {\mathbb{Q}}.\)

Then we may define the Euler characteristic in the usual way, except that #H\({}^ a\) is replaced by #(H\({}^ a_{tor})\), #H\({}^{a+2}\) is replaced by #(H\({}^{a+2}_{\cot or})=\#(D^{a+2}_{tor})\), and we add a regulator term \(R(G_ m)=\{\det <h_ i,d_ j>\}^{(- 1)^{a+1}}\), where \(\{h_ i\}\) and \(\{d_ j\}\) are bases of \(H^ a\) modulo torsion and \(D^{a+2}\) modulo torsion, respectively. In our case, \(a=1\), and \(H^ 3(X,{\mathbb{G}}_ m)\) ought to be, up to a finite group, the dual of the finitely generated abelian group consisting of 1-cycles modulo numerical equivalence.

In this paper, we show that, (1) (*) is true for X of dimension 0 or 1; (2) (*) is equivalent to Tate’s conjecture for X of dimension 2. In particular (*) is true if the \(\ell\)-part of \(H^ 2(X,{\mathbb{G}}_ m)\) is finite for any single prime \(\ell\), by the results of Tate and Milne; (3) (*) is true up to a power of p for X of any dimension, provided the \(\ell\)-part of \(H^ 2(X,{\mathbb{G}}_ m)\) is finite for any single prime \(\ell \neq p.\)

Along the way we show that \(H^ i(X,{\mathbb{G}}_ m)\) is finite for \(i\neq 1,2\), and 3, and that if in addition \(H^ 2(X,{\mathbb{G}}_ m)(\ell)\) is finite for one prime \(\ell \neq p\), we have \((i) H^ 2(X,{\mathbb{G}}_ m)(non-p)\) is finite, and (ii) \(H^ 3(X,{\mathbb{G}}_ m)\) is the dual of a finitely-generated abelian group (up to p-torsion).

Reviewer: I.G.Macdonald

### MSC:

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G15 | Finite ground fields in algebraic geometry |