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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.

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