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\(l\)-independence of the trace of monodromy. (English) Zbl 0980.14014
From the introduction: Let \(K\) be a complete discrete valuation field with finite residue field \(\mathbb{F}_{p^h}\), \(G_K\) the absolute Galois group of \(K\), \(I_K\) the inertia subgroup of \(G_K\) and \(W_K\) the Weil group of \(K\). We define a subset \(W^+_K\) of \(W_K\) to be \(W^+_K: =\{g\in W_K|u(g) \geq 0\}\). Let \(X\) be a variety over \(K\), \(\overline X\) means the scalar extension \(X\otimes_K \overline K\). We denote by \(l\) a prime number \(\neq p\). We consider the traces of the action of elements of \(W^+_K\) or \(W_K\) on the compact support étale cohomology \(\text{H}^i_c(\overline X,\mathbb{Q}_l) :={\underset {n} \varprojlim}\text{H}^i_c (\overline X,\mathbb{Z}/ l^n\mathbb{Z}) \bigotimes_{\mathbb{Z}_l} \mathbb{Q}_l\). Let us recall the following classical conjecture [J.-P. Serre and J. Tate, Ann. Math. (2) 88, 492-517 (1968; Zbl 0172.46101)]:
For any variety \(X\) over \(K\) and \(g\in W^+_K\), \(\text{Tr} (g^*;\text{H}^i_c(\overline X,\mathbb{Q}_l))\) is a rational integer which is independent of the choice of \(l\).
In this paper, we shall prove the following weak versions of the conjecture:
Proposition A. The trace \(\text{Tr} (g^*;\text{H}^i_c (\overline X,\mathbb{Q}_l))\) is an algebraic integer for any variety \(X\) over \(K\) and any \(g\in W^+_K\).
Theorem B. The alternating sum \(\sum^{2d}_{i=0} (-1)^i\text{Tr}(g^*;\text{H}^i_c (\overline X, \mathbb{Q}_l))\) is a rational integer which is independent of \(l\) for any variety \(X\) over \(K\) and any \(g\in W^+_K\).
When \(K\) is of mixed characteristics, we compare the traces for \(p\)-adic cohomologies and the traces for \(l\)-adic cohomologies in addition to the above result. Assume that \(X\) is proper and smooth over \(K\). The étale cohomology H\(^i(\overline X,\mathbb{Q}_p) :={\underset {n} \varprojlim}\text{H}^i(\overline X,\mathbb{Z}/ p^n\mathbb{Z}) \bigotimes_{\mathbb{Z}_p} \mathbb{Q}_p\) is a potentially semi-stable representation by the works of A. J. de Jong and T. Tsuji. Fontaine attached a \(p\)-adic representation of the Weil(-Deligne) group \(\widehat D_{\text{pst}} (V)\) to a \(p\)-adic potentially semi-stable representation \(V\). Thus we apply the functor \(\widehat D_{\text{pst}}\) on H\(^i (\overline X,\mathbb{Q}_p)\). Then \(\widehat D_{\text{pst}} (\text{H}^i(\overline X, \mathbb{Q}_p))\) gives a “good” \(p\)-adic representation of the Weil group \(W_K\) which completes the family of \(l\)-adic cohomologies for \(l\neq p\). We can prove:
Theorem D. For \(g\in W^+_K\) and a proper smooth variety \(X\) over \(K\), we have the following equality between rational integers: \[ \sum_i(-1)^i\text{Tr}\bigl( g^*;\text{H}^i(\overline X,\mathbb{Q}_l) \bigr)=\sum_i(-1)^i\text{Tr} \biggl(g^*; \widehat D_{\text{pst}} \bigl(\text{H}^i(\overline X,\mathbb{Q}_p) \bigr)\biggr). \] For the proof of theorem B, the result by A. J. de Jong concerning semi-stable reduction plays an essential role.

14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
14G15 Finite ground fields in algebraic geometry