Let \(K\) be a finite extension of \({\mathbb Q}_p\) or of \({\mathbb F}_{p}((T))\) (\(p\) prime), and choose a separable algebraic closure \(\overline K\) of \(K\). The residue field \(\overline F\) of \(\overline K\) is an algebraic closure of the residue field \(F\) of \(K\), and there is canonical surjection \(\rho:G_K\rightarrow G_F\), with \(G_K=\text{ Gal}(\overline K|K)\) and \(G_F=\text{ Gal}(\overline F|F)\); the latter is generated as a profinite group by \(\text{ Fr}\), the inverse of the element \(a\mapsto a^q\), with \(q=\text{ Card}F\); the kernel of \(\rho\) is the inertia subgroup \(I_K\) of \(G_K\). The Weil group \(W_K\) is the inverse image in \(G_K\) of the (dense) subgroup of \(G_F\) generated by \(\text{ Fr}\) ; it comes with a homomorphism \(n:W_K\rightarrow{\mathbb Z}\), defined by \(\rho(\sigma)=\text{ Fr}^{n(\sigma)}\) for all \(\sigma\in W_K\).
Let \(X\) be a \(K\)-scheme of finite type, and put \(\overline X=X\times_K\overline K\). By functoriality, every \(\sigma\in W_K\) gives rise to an automorphism \(\sigma_*\) of \(H^r(\overline X,{\mathbb Q}_l)\). For smooth, proper, purely \(d\)-dimensional \(X\), every correspondence \(\Gamma\) on \(X\), i.e. every \(d\)-cycle \(\Gamma\) on \(X\times_K X\) modulo rational equivalence, gives rise to the endomorphism \(\Gamma^*={\text{ pr}_1}_* \circ([\Gamma]\smile\phantom{\Gamma})\circ{\text{ pr}_2}^*\) of \(H^r(\overline X,{\mathbb Q}_l)\).
Let \(X\) be as above and assume that \(l\neq p\). For every \(\sigma\in W_K\) and every correspondence \(\Gamma\) on \(X\), the alternating sum \[ \sum_{r=0}^{2d}(-1)^r\,\text{ Tr}\left(\Gamma^*\circ\sigma_*:H^r(\overline X,{\mathbb Q}_l)\right) \] – a priori an element of \({\mathbb Q}_l\) — belongs to \(q^{d\inf(0,n(\sigma))}{\mathbb Z}\) and is independent of \(l\) (Theorem 0.1). When \(\Gamma\) is the diagonal, this was proved by T. Ochiai [Math. Ann. 315, No. 2, 321–340 (1999; Zbl 0980.14014)]; when \(\sigma\) belongs to the wild inertia, by the author and K. Kato [Publ. Math., Inst. Hautes Étud. Sci. 100, 5–151 (2004; Zbl 1099.14009)].
For \(X\) an abelian variety, the trace of \(\Gamma^*\circ\sigma_*:H^r(\overline X,{\mathbb Q}_l)\) was shown to be independent of \(l\) by Grothendieck. The author shows that if \(X\) is a curve or a surface, the trace of \(\Gamma^*\circ\sigma_*:H^r(\overline X,{\mathbb Q}_l)\) belongs to \(q^{d\inf(0,n(\sigma))}{\mathbb Z}\) and is independent of \(l\) (Corollary 0.2). Further, if \(n(\sigma)\geq0\), the characteristic polynomials of \(\sigma_*:H^r(\overline X,{\mathbb Q}_l)\) and of \(\text{ Fr}_*:H^r(\overline X,{\mathbb Q}_l)^{I_K}\) have coefficients in \({\mathbb Z}\) and are independent of \(l\). Moreover, the complex roots of the latter polynomial have absolute value \(q^{s/2}\), for a fixed \(s\) between \(0\) and \(r\) (Corollary 0.4). These results confirm, for \(d=1\), \(2\), conjectures of J.-P. Serre [Facteurs locaux des fonctions zeta des variétés algébriques, Theorie Nombres, Semin. Delange-Pisot-Poitou 11 (1969/70), No. 19 (1970; Zbl 0214.48403)] for arbitrary dimensions \(d\).
The main result is deduced from properties of the weight spectral sequence of Rapoport-Zink for semistable schemes over the ring of integers of \(K\), using de Jong alterations and the Lefschetz trace formula. The author expects that the same argument, carried out for the weight spectral sequence of Mokrane, should give \[ \text{ Tr}\left(\Gamma^*\circ\sigma_*:H^r(\overline X,{\mathbb Q}_l)\right) = \text{ Tr}\left(\Gamma^*\circ\sigma_*:DH^r(\overline X,{\mathbb Q}_p)\right), \] in the mixed charateristic case, where \(DH^r(\overline X,{\mathbb Q}_p)\) is the module functorially associated by Fontaine to the potentially semistable \({\mathbb Q}_p\)-representation \(H^r(\overline X,{\mathbb Q}_p)\) of \(G_K\). When \(\Gamma\) is the diagonal, this has been established by Ochiai [loc. cit.].
Similar results for rigid \(K\)-spaces have recently been obtained by Y. Mieda [math.NT/0508509, math.AG/0601100], by a modification of the author’s method.