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**On the existence of \(F\)-crystals.**
*(English)*
Zbl 1126.14023

For a prime number \(p\), denote by \(W(\overline\mathbb{F}_p)\) the ring of Witt vectors of an algebraic closure \(\overline\mathbb{F}_p\) of the finite field \(\mathbb{F}_p=\mathbb{Z}/p \mathbb{Z}\). Let \(L\) be the fraction field of \(W(\overline \mathbb{F}_p)\) and consider the relative Frobenius automorphism \(\sigma\in \operatorname{Aut}(L/\mathbb{Q}_p)\). By definition, an \(F\)-isocrystal with respect to the prime number \(p\) is a pair \((N,F)\) consisting of a finite-dimensional vector space \(N\) over \(L\) and a \(\sigma\)-linear bijective endomorphism \(F\) of \(N\), and these \(F\)-isocrystals form a category in an obvious way.

According to Dieudonné’s theory, each \(F\)-isocrystal \((N,F)\) of dimension \(n=\dim_L(N)\) defines an element \(\nu(N,F)=(\nu_1,\dots, \nu_n)\) in \((\mathbb{Q}^n)_+:=\{(r_1,\dots,r_n)\in\mathbb{Q}^n;r_1\geq r_2\geq \cdots\geq r_n\}\), its so-called Newton slope sequence, and the \(n\)-dimensional \(F\)-isocrystals are classified up to isomorphism by their Newton slope sequence. Now, if \(M\) is a \(W(\mathbb{F}_p)\)-lattice in \(N\), then the relative position of \(M\) and \(F(M)\) is measured by the so-called Hodge slope sequence \(\mu(M):=\text{inv}(M,F(M))\in (\mathbb{Z}^n)_+= \mathbb{Z}^n\cap(\mathbb{Q}^n)_+\), where \(\mu(M)=(\mu_1,\dots,\mu_n)\) if and only if there exists a \(W(\overline\mathbb{F}_p)\)-basis \(e_1,\dots,e_n\) of \(M\) such that \(p^{\mu_1}e_1,\dots,p^{\mu_n}e_n\) is a \(W(\overline \mathbb{F}_p)\)-basis of the lattice \(F(M)\). By an earlier theorem of B. Mazur, the Newton and the Hodge slope sequences are related by the inequality \(\mu(M)\geq\nu(N,F)\) in \((\mathbb{Q}^n)_+\).

Mazur’s inequality is the starting point of the paper under review, in which the authors prove a certain converse statement. Namely, one of their main results (Theorem A) asserts the following: Let \((N,F)\) be an isocrystal of dimension \(n\) over \(\overline\mathbb{F}_p\), and let \(\mu\in(\mathbb{Z}^n)_+\) such that \(\mu\geq\nu (N,F)\) in \((\mathbb{Q}^n)_+\). Then there exists a \(W(\overline \mathbb{F}_p)\)-lattice \(M\) in \(N\) satisfying \(\mu(M)=\mu\).

As the authors point out, this converse of Mazur’s theorem may be considered as a statement on generalized affine Deligne-Lusztig varieties. Actually, this viewpoint provides the methodological framework for the proof of Theorem A. More precisely, the other main result of the present paper (Theorem B) ensures that certain Deligne-Lusztig sets associated with the group \(\text{GL}_n\) (or \(\text{GS}p_{2n}\), respectively) are non-empty.

Using Theorem B, the converse to Mazur’s inequality is proved by strengthening an earlier result of M. Rapoport [Manuscr. Math. 101, 153–166 (2000; Zbl 0941.22006)] with respect to the groups \(\text{GL}_n\) and \(\text{GSP}_{2n}\). In the body of the paper, the authors deal more generally with a finite extension of \(\mathbb{Q}_n\) and with the completion \(L\) of its maximal unramified extension, that is with general \(F\)-isocrystals.

Finally, the problem of generalizing the foregoing results to other unramified reductive groups is discussed. This leads to a precisely formulated conjecture on the non-emptiness of certain Deligne-Lusztig varieties (Conjecture 3.1) in the general case.

As to the ultimate motivation for the results proved in the present paper, the authors mention that these allow to reformulate, in many cases, the Langlands-Rapoport conjecture on the reduction of Shimura varieties [cf. R. Langlands and M. Rapoport, J. Reine Angew. Math. 378, 113–220 (1987; Zbl 0615.14014)].

According to Dieudonné’s theory, each \(F\)-isocrystal \((N,F)\) of dimension \(n=\dim_L(N)\) defines an element \(\nu(N,F)=(\nu_1,\dots, \nu_n)\) in \((\mathbb{Q}^n)_+:=\{(r_1,\dots,r_n)\in\mathbb{Q}^n;r_1\geq r_2\geq \cdots\geq r_n\}\), its so-called Newton slope sequence, and the \(n\)-dimensional \(F\)-isocrystals are classified up to isomorphism by their Newton slope sequence. Now, if \(M\) is a \(W(\mathbb{F}_p)\)-lattice in \(N\), then the relative position of \(M\) and \(F(M)\) is measured by the so-called Hodge slope sequence \(\mu(M):=\text{inv}(M,F(M))\in (\mathbb{Z}^n)_+= \mathbb{Z}^n\cap(\mathbb{Q}^n)_+\), where \(\mu(M)=(\mu_1,\dots,\mu_n)\) if and only if there exists a \(W(\overline\mathbb{F}_p)\)-basis \(e_1,\dots,e_n\) of \(M\) such that \(p^{\mu_1}e_1,\dots,p^{\mu_n}e_n\) is a \(W(\overline \mathbb{F}_p)\)-basis of the lattice \(F(M)\). By an earlier theorem of B. Mazur, the Newton and the Hodge slope sequences are related by the inequality \(\mu(M)\geq\nu(N,F)\) in \((\mathbb{Q}^n)_+\).

Mazur’s inequality is the starting point of the paper under review, in which the authors prove a certain converse statement. Namely, one of their main results (Theorem A) asserts the following: Let \((N,F)\) be an isocrystal of dimension \(n\) over \(\overline\mathbb{F}_p\), and let \(\mu\in(\mathbb{Z}^n)_+\) such that \(\mu\geq\nu (N,F)\) in \((\mathbb{Q}^n)_+\). Then there exists a \(W(\overline \mathbb{F}_p)\)-lattice \(M\) in \(N\) satisfying \(\mu(M)=\mu\).

As the authors point out, this converse of Mazur’s theorem may be considered as a statement on generalized affine Deligne-Lusztig varieties. Actually, this viewpoint provides the methodological framework for the proof of Theorem A. More precisely, the other main result of the present paper (Theorem B) ensures that certain Deligne-Lusztig sets associated with the group \(\text{GL}_n\) (or \(\text{GS}p_{2n}\), respectively) are non-empty.

Using Theorem B, the converse to Mazur’s inequality is proved by strengthening an earlier result of M. Rapoport [Manuscr. Math. 101, 153–166 (2000; Zbl 0941.22006)] with respect to the groups \(\text{GL}_n\) and \(\text{GSP}_{2n}\). In the body of the paper, the authors deal more generally with a finite extension of \(\mathbb{Q}_n\) and with the completion \(L\) of its maximal unramified extension, that is with general \(F\)-isocrystals.

Finally, the problem of generalizing the foregoing results to other unramified reductive groups is discussed. This leads to a precisely formulated conjecture on the non-emptiness of certain Deligne-Lusztig varieties (Conjecture 3.1) in the general case.

As to the ultimate motivation for the results proved in the present paper, the authors mention that these allow to reformulate, in many cases, the Langlands-Rapoport conjecture on the reduction of Shimura varieties [cf. R. Langlands and M. Rapoport, J. Reine Angew. Math. 378, 113–220 (1987; Zbl 0615.14014)].

Reviewer: Werner Kleinert (Berlin)