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\(\ell \)-independence of the trace of local monodromy in a relative case (with an appendix by Qing Lu and Weizhe Zheng)

Abstract

For a family of varieties, we prove that the alternating sum of the traces of “local” monodromy acting on the \(\ell \)-adic étale cohomology groups of the generic fiber is an integer that is independent of \(\ell \). In the course of the proof, we also establish a result on fixed points.

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Notes

  1. 1.

    The constructible topology on a noetherian scheme is the topology generated by Zariski open subsets and Zariski closed subsets.

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Acknowledgements

The author would like to thank his advisor T. Saito for helpful discussions, reading this article carefully, and giving comments which improved various points of the article. He also thanks D. Takeuchi for helpful discussions and Q. Lu and W. Zheng for sharing another proof of Theorem 1.1 and kindly suggesting the author include it in the “Appendix”. Finally, he thanks the referee for reading this article carefully and giving helpful comments. The research was supported by the Program for Leading Graduate Schools, MEXT, Japan and also by JSPS KAKENHI Grant No. 18J12981.

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Appendix: \(\ell \)-independence over Henselian valuation fields (by Qing Lu and Weizhe Zheng)

Appendix: \(\ell \)-independence over Henselian valuation fields (by Qing Lu and Weizhe Zheng)

In this “Appendix” we prove results on \(\ell \)-independence and integrality of \(\ell \)-adic cohomology over Henselian valuation fields with not necessarily discrete valuations, for the action of inertia and Weil subgroups of the Galois group. We deduce these results from relative results on compatible systems and integral sheaves over discrete valuation fields ([7, 25]) using the valuative criteria of [5]. Our result for inertia action (Theorem A.1) slightly generalizes Hiroki Kato’s Theorem 6.1 and our method is different from his. We thank him for allowing us to present our results in this “Appendix”.

Let K be a field equipped with an action of a finite group G. Let X be an algebraic space of dimension d of finite presentation over K, equipped with an equivariant action of G. Let \(K_0=K^G\), \(H=\mathrm {Gal}(K/K_0)\) and let \({\bar{K}}\) denote a separable closure of K. Let \(X_{\bar{K}}=X\otimes _K {\bar{K}}\). For every prime number \(\ell \) invertible in K and every integer i, \(G\times _H \mathrm {Gal}({\bar{K}}/K_0)\) acts continuously on \(H^i_c(X_{{\bar{K}}},\mathbb {Q}_\ell )\) and \(H^i(X_{{\bar{K}}},\mathbb {Q}_\ell )\).

Theorem A.1

Let \(\mathbb {L}\) be a set of prime numbers not containing the characteristic of K. Then for every \((g,\sigma )\in G\times _H E_{K_0/\mathrm {Spec}(\mathbb {Z}[{\mathbb {L}}^{-1}])}\) and every i, the eigenvalues of \((g,\sigma )\) acting on \(H^i_c(X_{\bar{K}},\mathbb {Q}_\ell )\) and \(H^i(X_{{\bar{K}}},\mathbb {Q}_\ell )\) are roots of unity for \(\ell \in \mathbb {L}\), and

$$\begin{aligned} \sum _i (-1)^i\mathrm {tr}((g,\sigma ), H^i_c(X_{{\bar{K}}},\mathbb {Q}_\ell ))=\sum _i (-1)^i\mathrm {tr}((g,\sigma ), H^i(X_{{\bar{K}}},\mathbb {Q}_\ell )) \end{aligned}$$

is a rational integer independent of \(\ell \in \mathbb {L}\).

Recall from Definition 2.1 that \(E_{K_0/\mathrm {Spec}(\mathbb {Z}[{\mathbb {L}}^{-1}])}\subseteq \mathrm {Gal}({\bar{K}}/K_0)\) denotes the ramified part, namely the closure of the union of the images of \(\mathrm {Gal}({\bar{L}}/L)\rightarrow \mathrm {Gal}({\bar{K}}/K_0)\) for all commutative diagrams

with \(\mathcal {O}_{L}\) a strictly Henselian valuation ring of fraction field L, and \({\bar{L}}\) a separable closure of L. In particular, if \(K_0\) is the fraction field of a Henselian valuation ring of residue characteristic \(\not \in \mathbb {L}\), then \(E_{K_0/\mathrm {Spec}(\mathbb {Z}[{\mathbb {L}}^{-1}])}\) is the inertia subgroup of \(\mathrm {Gal}(\bar{K}/K_0)\).

Theorem A.2

Assume that \(K_0\) is the fraction field of a Henselian valuation ring \(\mathcal {O}\) of finite residue field \(\mathbb {F}_q\). Then for all \((g,\sigma )\in G\times _{H} W({\bar{K}}/K_0)\) with \(\sigma \) of degree \(\nu \ge 0\), \(i\in \mathbb {Z}\), \(\ell \not \mid q\), and all eigenvalues \(\alpha \) of \((g,\sigma )\) acting on \(H^i_c(X_{\bar{K}},\mathbb {Q}_\ell )\) and \(H^i(X_{{\bar{K}}},\mathbb {Q}_\ell )\), the numbers \(\alpha \), \(q^{ \nu (d-i)}\alpha \), \(q^{\nu d}\alpha ^{-1}\), \(q^{\nu i}\alpha ^{-1}\) are algebraic integers, and

$$\begin{aligned} \sum _i (-1)^i\mathrm {tr}((g,\sigma ), H^i_c(X_{{\bar{K}}},\mathbb {Q}_\ell )),\quad \sum _i (-1)^i\mathrm {tr}((g,\sigma ), H^i(X_{{\bar{K}}},\mathbb {Q}_\ell )) \end{aligned}$$

are rational integers independent of \(\ell \ne q\). Here \(W({\bar{K}}/K_0)\) denotes the inverse image of the Weil group \(W(\overline{\mathbb {F}_q}/\mathbb {F}_q)\) under the reduction map \(r:\mathrm {Gal}({\bar{K}}/K_0)\rightarrow \mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)\), and that \(\sigma \) has degree \(\nu \) means \(r(\sigma )=\mathrm {Fr}_q^\nu \), where \(\mathrm {Fr}_q\) denotes the geometric Frobenius \(a\mapsto a^{1/q}\).

Theorem A.2 was previously known under the additional assumption that \(\mathcal {O}\) is a finite field or a discrete valuation ring ([1, 26, 7, 25, 27, Appendix]).

Proof of the theorems

In Theorem A.1, we may assume by continuity (see Lemma A.3 below) that \(K_0=L\) is the fraction field of a strictly Henselian valuation ring \(\mathcal {O}\) over \(\mathbb {Z}[{\mathbb {L}}^{-1}]\). In this case, the equality follows from the equivariant form of a theorem of Laumon [22, Theorem 2.2]. We will prove the rest of Theorem A.1 and Theorem A.2 in parallel. In Theorem A.2 we take \(\mathbb {L}\) to be the set of primes \(\ell \not \mid q\).

Let us prove that the alternating sums in the theorems are in \(\mathbb {Q}\) and independent of \(\ell \in \mathbb {L}\). The first step is to take care of the finite group action. Up to replacing G by \(\langle g\rangle \), we may assume that G is abelian. We fix \(\sigma \), of image \({\bar{\sigma }}\in H\). Let \(t_{g,\ell }^c\) and \(t_{g,\ell }\) denote the alternating sums in the theorems. Let \({\bar{\mathbb {Q}}}\) be an algebraic closure of \(\mathbb {Q}\) and let I be the set of pairs \((\ell ,\iota )\), where \(\ell \in \mathbb {L}\) and \(\iota :{\bar{\mathbb {Q}}}\rightarrow \overline{\mathbb {Q}_\ell }\) is an embedding. It suffices to show the existence of \(t^{(c)}_{g}\in {\bar{\mathbb {Q}}}\) such that \(\iota (t^{(c)}_{g})=t^{(c)}_{g,\ell }\) for all \((\ell ,\iota )\in I\) (cf. [7, Remarque 1.18 (iv)]). For any character \(\chi :G\rightarrow ({\bar{\mathbb {Q}}})^\times \), let \(V_{\iota \chi }\) denote the one-dimensional \(\overline{\mathbb {Q}_\ell }\)-representation of character \(\iota \chi \) and let \(\mathcal {L}_{\iota \chi }\) denote the corresponding lisse \(\overline{\mathbb {Q}_\ell }\)-sheaf on the quotient stack [X/G]. Let \([X/G]_{{\bar{K}}}:=[X/G]\otimes _{K_0} {\bar{K}}\) and let \(\pi :G\rightarrow H\). Then \(H^*_{(c)}([X/G]_{{\bar{K}}},\mathcal {L}_{\iota \chi })\simeq (H^*_{(c)}(X_{{\bar{K}}},\overline{\mathbb {Q}_\ell })\otimes _{\overline{\mathbb {Q}_\ell }} V_{\iota \chi })^{\mathrm {Ker}(\pi )}\), so that

$$\begin{aligned} \mathrm {tr}\big (\sigma ,H^*_{(c)}\big ([X/G]_{{\bar{K}}},\mathcal {L}_{\iota \chi }\big )\big ) =\frac{1}{\#\mathrm {Ker}(\pi )}\sum _{g\in \pi ^{-1}({\bar{\sigma }})}t^{(c)}_{g,\ell }\iota (\chi (g)). \end{aligned}$$

Since every function \(\pi ^{-1}({\bar{\sigma }})\rightarrow {\bar{\mathbb {Q}}}\), in particular every indicator function, is a \({\bar{\mathbb {Q}}}\)-linear combination of the \(\chi |_{\pi ^{-1}({\bar{\sigma }})}\)’s, it suffices to show that \((\mathrm {tr}(\sigma ,H^*_{(c)}([X/G]_{{\bar{K}}},\mathcal {L}_{\iota \chi })))\) is I-compatible.

The rest of the proof is similar to the proof of [5, Theorem 1.4], except that the base here may have mixed characteristics. By standard limit arguments, there exists a finitely generated sub-algebra \(R\subseteq K_0\) over \(\mathbb {Z}\) such that X and the action of G are defined over \(B=\mathrm {Spec}(R)\): there exists an algebraic space \(\mathcal {X}\) of finite presentation over B, equipped with an action of G by B-automorphisms, such that we have a G-equivariant isomorphism \(X\simeq \mathcal {X}\times _B \mathrm {Spec}(K)\) over K. Let \(f:[\mathcal {X}/G]\rightarrow B\). If the residue field of \(\mathcal {O}\) has characteristic \(>0\), let p be its characteristic. Otherwise (this case may only occur in Theorem 1), noting that it suffices to show J-compatibility for every finite subset \(J\subseteq I\), we may assume that \(\mathbb {L}\) is not the set of all primes and we choose \(p\not \in \mathbb {L}\). In both cases, we obtain a commutative square

where \(\mathbb {Z}_{(p)}\) denotes the Henselization of \(\mathbb {Z}\) at the ideal (p), which is an excellent Henselian discrete valuation ring, and \(B_{(p)}=\mathrm {Spec}(R\otimes _\mathbb {Z}\mathbb {Z}_{(p)})\). Let \(h:[\mathcal {X}_{(p)}/G]\rightarrow B_{(p)}\) denote the base change of f to \(B_{(p)}\). We still denote by \(\mathcal {L}_{\iota \chi }\) the lisse \(\overline{\mathbb {Q}_\ell }\)-sheaf on \([\mathcal {X}_{(p)}/G]\) given by \(V_{\iota \chi }\). By [7] (as summarized in [5, Theorem 2.3]), \((R h_{!}\mathcal {L}_{\iota \chi })\) and \((R h_{*}\mathcal {L}_{\iota \chi })\) are I-compatible systems on \(B_{(p)}\). Moreover, since \(\mathrm {Spec}(K_0)\) maps to the generic point of \(B_{(p)}\), \(H^i([X/G]_{{\bar{K}}},\mathcal {L}_{\iota \chi })\simeq (R^{i}h_{*}\mathcal {L}_{\iota \chi })_{{\bar{K}}}\) and similarly for \(H^i_c\). Thus by the valuative criterion for compatible systems [5, Corollary 1.3], \((\mathrm {tr}(\sigma ,H^*_{(c)}([X/G]_{{\bar{K}}},\mathcal {L}_{\iota \chi })))\) is I-compatible. This finishes the proof of rationality and \(\ell \)-independence.

It remains to show the assertions on eigenvalues. For these assertions we may replace \((g,\sigma )\) by \((g,\sigma )^n\) for any \(n\ge 1\). Thus we may assume \(G=\{1\}\). We conclude the proof of Theorem A.1 by the last assertion of [5, Corollary 3.10] applied to \(R^{i}h_{!}\overline{\mathbb {Q}_\ell }\) and \(R^{i}h_{*}\overline{\mathbb {Q}_\ell }\).

Finally, we prove the assertion on eigenvalues in Theorem A.2. If \(K_0\) has characteristic 0, we may remove the fiber at p from B. Thus we may assume that \(B_{(p)}\) is above either the closed point or the generic point of \(\mathrm {Spec}(\mathbb {Z}_{(p)})\). Further shrinking B if necessary, we may assume that \(B_{(p)}\) is regular of pure dimension \(d'\), \(\mathcal {X}_{(p)}\) has dimension \(d'':=d+d'\), and \(R^{i}h_{!}\overline{\mathbb {Q}_\ell }\) and \(R^{i}h_{*}\overline{\mathbb {Q}_\ell }\) are lisse. By [25, Proposition 6.4], \({{R^{i}}{h_{!}}\overline{\mathbb {Q}_\ell }}\), \({{R^{i}}{h_{!}}\overline{\mathbb {Q}_\ell }(i-d)}\), \(({{R^{i}}{h_{!}}\overline{\mathbb {Q}_\ell }(i)})^{\vee }\), \(({{R^{i}}{h_{!}}\overline{\mathbb {Q}_\ell }(d)})^{\vee }\), \({{R^{i}}{h_{*}}\overline{\mathbb {Q}_\ell }}\), \((R^{i}h_{*}\overline{\mathbb {Q}_\ell }(i))^{\vee }\) are integral sheaves on \(B_{(p)}\). Here \(\mathcal {F}^{\vee }:=\mathcal {H} om (\mathcal {F},\overline{\mathbb {Q}_\ell })\) for \(\mathcal {F}\) lisse. Moreover, \({Rh}_{*}\overline{\mathbb {Q}_\ell }\simeq R\mathcal {H} om (Rh_{!}D_{\mathcal {X}_{(p)}}\overline{\mathbb {Q}_\ell },\overline{\mathbb {Q}_\ell }(d')[2d'])\), where we adopted the usual normalization of dualizing functor \(D_{Y}:=R\mathcal {H} om (-,Ra^{!}\overline{\mathbb {Q}_\ell })\) for \(a:Y\rightarrow \mathrm {Spec}(k)\). By [25, Propositions 6.2, 6.4], \((R^{i}h_{!}D_{\mathcal {X}_{(p)}}\overline{\mathbb {Q}_\ell })(-d'')\) and \((R^{i}h_{!}D_{\mathcal {X}_{(p)}}\overline{\mathbb {Q}_\ell }(i+d''))^{\vee }\) are integral. It follows that \(R^{i}h_{*}\overline{\mathbb {Q}_\ell }(i-d)\) and \((R^{i}h_{*}\overline{\mathbb {Q}_\ell }(d))^{\vee }\) are integral. We conclude by the valuative criterion for integrality [5, Corollary 3.10]. \(\square \)

The following continuity lemma is extracted from the proof of [5, Lemma 4.5]. Let \(P_\ell ^{n}\simeq \overline{\mathbb {Q}_\ell }^n\) denote the space of monic polynomials in \(\overline{\mathbb {Q}_\ell }[T]\) of degree n.

Lemma A.3

Let I be a finite set. For each i, let \(\ell _i\) be a prime and \(n_i\ge 0\) an integer. Let E be a topological space equipped with continuous maps \(\rho _{i}:E\rightarrow \mathrm {GL}_{{n_{{i}}}}(\overline{\mathbb {Q}_{\ell _i}})\) and let \(U\subseteq E\) be a dense subspace. Let \(S\subseteq \prod _i P_{{\ell _{i}}}^{{n_{i}}}\) be a subset. Assume that \(\rho _i(\sigma )\) is quasi-unipotent and \((\det (T\cdot 1-\rho _i(\sigma )))_i\in S\) for all \(\sigma \in U\). Then the same holds for all \(\sigma \in E\).

Proof

Consider the continuous map \(\lambda :E\rightarrow \prod _i P_{{\ell _{i}}}^{n_{i}}\) carrying \(\sigma \) to \((\det (\rho _i(\sigma )-T\cdot 1))_i\). As in [5, Remark 2.11], let \({P_{\ell }^{n,\mathrm {qu}}}\subseteq {P_{\ell }^{n}}\) denote the subset of polynomials whose roots are all roots of unity, which is discrete and closed. By assumption, \(\lambda (U)\subseteq S\cap \prod _{i} P_{\ell _{i}}^{n_{i},\mathrm {qu}}\). Thus \(\lambda (U)=\lambda (E)\). \(\square \)

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Kato, H. \(\ell \)-independence of the trace of local monodromy in a relative case (with an appendix by Qing Lu and Weizhe Zheng). manuscripta math. 166, 287–314 (2021). https://doi.org/10.1007/s00229-020-01234-x

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Mathematics Subject Classification

  • 11G25
  • 14D10
  • 14G20