Local \(\mathbb{P}\)-shtukas and their relation to global \({\mathfrak s}\)-shtukas.

*(English)*Zbl 1348.14110This is a highly interesting paper on parameterization questions for local \(\mathbb{P}\)-shtukas and for global \(\mathfrak{G}\)-shtukas. It continues works of V. G. Drinfel’d [Funct. Anal. Appl. 21, No. 1–3, 107–122 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 23–41 (1987; Zbl 0665.12013)], L. Lafforgue [Invent. Math. 147, No. 1, 1–241 (2002; Zbl 1038.11075)], G. Faltings [J. Eur. Math. Soc. (JEMS) 5, No. 1, 41–68 (2003; Zbl 1020.14002)], U. Hartl [Prog. Math. 239, 167–222 (2005; Zbl 1137.11322)] and U. Hartl and E. Viehmann [J. Reine Angew. Math. 656, 87–129 (2011; Zbl 1225.14036)].

Recall that local \(\mathbb{P}\)-shtukas are the functional field analogs of \(p\)-divisible groups with additional structure and moduli stacks of global \(\mathfrak{G}\)-shtukas are the functional field analogs for Shimura varieties. Here \(\mathbb{P}\) is a flat affine group scheme of finite type over \(\mathrm{Spec }\mathbb{F}[[z]]\) and \(\mathfrak{G}\) is a flat affine group scheme of finite type over a smooth projective geometrically irreducible curve over \(\mathbb{F}_{q}\); authors let \(\mathbb{F}\) be a finite field extension of a finite field \(\mathbb{F}_{q}\) with \(q\) elements and characteristic \(p\).

The authors main theme is the relation between global \(\mathfrak{G}\)-shtukas and local \(\mathbb{P}\)-shtukas and authors of the paper under review prove the analog of a theorem of Serre and Tate over functional fields ‘stating the equivalence between the deformations of a global \(\mathfrak{G}\)-shtuka and its associated local \(\mathbb{P}\)-shtukas‘ . Results explaining relation between local \(\mathbb{P}\)-shtukas and Galois representations as well as results on the existence of spaces by M. Rapoport and Th. Zink [Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] and their use to uniformize the moduli stack of global \(\mathfrak{G}\)-shtukas also presented.

Similar partial results have appeared in dissertation by R. K. Singh [Local shtukas and divisible local Anderson-modules. Münster: Univ. Münster, Fachbereich Mathematik und Informatik (Diss.) (2012; Zbl 1262.14001)] and in dissertation by M. E. A. Rad [Uniformizing the moduli stacks of global \(\mathfrak G\)-shtukas. Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss.) (2012; Zbl 1298.14005)] both written under U. Hartl (Münster).

The generalization of purity of the Newton stratification in the context of local \(G\)-shtukas with applications to a generalization of Grothendieck’s conjecture on deformations of \(p\)-divisible groups with given Newton polygons were obtained by E. Viehmann [Am. J. Math. 135, No. 2, 499–518 (2013; Zbl 1278.14062)].

The range of topics of the paper under review is indicated by the titles of the sections: 1. Introduction; 2. Local \(\mathbb{P}\)-shtukas and global \(\mathfrak{G}\)-shtukas; 3. Tate modules for local \(\mathbb{P}\)-shtukas; 4. The Rapoport-Zink spaces for local \(\mathbb{P}\)-shtukas; 5. The relation between global \(\mathfrak{G}\)-shtukas and local \(\mathbb{P}\)-shtukas.

Section 1 comprises main results and list of notation and conventions. In section 2, elements of the theory of loop groups, of local and global shtukas are introduced. These include formal torsor, local \(\mathbb{P}\)-shtukas, groupoid of local \(\mathbb{P}\)-shtukas, quasi-isogeny between two local \(\mathbb{P}\)-shtukas, global \(\mathfrak{G}\)-shtuka. Main results include Proposition 2.4 on a canonical equivalence between the category fibered in groupoids that assigne to each \(\mathbb{F}\)-scheme \(S\) the groupoid consisting of all formal torsors (the generalization of results by U. Hartl and E. Viehmann [J. Reine Angew. Math. 656, 87–129 (2011; Zbl 1225.14036)]) and Proposition 2.11 on rigidity of quasi-isogenies for local \(\mathbb{P}\)-shtukas. Section 3 concerns the relation between local \(\mathbb{P}\)-shtukas and Galois representations which is given by the associated Tate module. The (dual) Tate functor and the rational (dual) Tate functor are defined. Under certain conditions (dual) Tate functors are equivalences between category of étale local shtukas over \(S\) to the category of finite free \(\mathbb{F}[[z]]\)-modules equipped with continuous action of the algebraic fundamental group of \(S\) at its geometric point. Section 4 works out unbounded Rapoport-Zink spaces for local \(\mathbb{P}\)-shtukas. Boundedness conditions for local \(\mathbb{P}\)-shtukas and bounded local \(\mathbb{P}\)-shtukas also introduced and investigated. Representability of the bounded Rapoport-Zink functor is proved. The results are too technical to be stated here in details. They are applied later to uniformize the moduli stack of global \(\mathfrak{G}\)-shtukas. The authors conclude this impressive work by proving the rigidity of quasi-isogenies for global \(\mathfrak{G}\)-shtukas and by proving the analog of the Serre-Tate theorem.

Recall that local \(\mathbb{P}\)-shtukas are the functional field analogs of \(p\)-divisible groups with additional structure and moduli stacks of global \(\mathfrak{G}\)-shtukas are the functional field analogs for Shimura varieties. Here \(\mathbb{P}\) is a flat affine group scheme of finite type over \(\mathrm{Spec }\mathbb{F}[[z]]\) and \(\mathfrak{G}\) is a flat affine group scheme of finite type over a smooth projective geometrically irreducible curve over \(\mathbb{F}_{q}\); authors let \(\mathbb{F}\) be a finite field extension of a finite field \(\mathbb{F}_{q}\) with \(q\) elements and characteristic \(p\).

The authors main theme is the relation between global \(\mathfrak{G}\)-shtukas and local \(\mathbb{P}\)-shtukas and authors of the paper under review prove the analog of a theorem of Serre and Tate over functional fields ‘stating the equivalence between the deformations of a global \(\mathfrak{G}\)-shtuka and its associated local \(\mathbb{P}\)-shtukas‘ . Results explaining relation between local \(\mathbb{P}\)-shtukas and Galois representations as well as results on the existence of spaces by M. Rapoport and Th. Zink [Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] and their use to uniformize the moduli stack of global \(\mathfrak{G}\)-shtukas also presented.

Similar partial results have appeared in dissertation by R. K. Singh [Local shtukas and divisible local Anderson-modules. Münster: Univ. Münster, Fachbereich Mathematik und Informatik (Diss.) (2012; Zbl 1262.14001)] and in dissertation by M. E. A. Rad [Uniformizing the moduli stacks of global \(\mathfrak G\)-shtukas. Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss.) (2012; Zbl 1298.14005)] both written under U. Hartl (Münster).

The generalization of purity of the Newton stratification in the context of local \(G\)-shtukas with applications to a generalization of Grothendieck’s conjecture on deformations of \(p\)-divisible groups with given Newton polygons were obtained by E. Viehmann [Am. J. Math. 135, No. 2, 499–518 (2013; Zbl 1278.14062)].

The range of topics of the paper under review is indicated by the titles of the sections: 1. Introduction; 2. Local \(\mathbb{P}\)-shtukas and global \(\mathfrak{G}\)-shtukas; 3. Tate modules for local \(\mathbb{P}\)-shtukas; 4. The Rapoport-Zink spaces for local \(\mathbb{P}\)-shtukas; 5. The relation between global \(\mathfrak{G}\)-shtukas and local \(\mathbb{P}\)-shtukas.

Section 1 comprises main results and list of notation and conventions. In section 2, elements of the theory of loop groups, of local and global shtukas are introduced. These include formal torsor, local \(\mathbb{P}\)-shtukas, groupoid of local \(\mathbb{P}\)-shtukas, quasi-isogeny between two local \(\mathbb{P}\)-shtukas, global \(\mathfrak{G}\)-shtuka. Main results include Proposition 2.4 on a canonical equivalence between the category fibered in groupoids that assigne to each \(\mathbb{F}\)-scheme \(S\) the groupoid consisting of all formal torsors (the generalization of results by U. Hartl and E. Viehmann [J. Reine Angew. Math. 656, 87–129 (2011; Zbl 1225.14036)]) and Proposition 2.11 on rigidity of quasi-isogenies for local \(\mathbb{P}\)-shtukas. Section 3 concerns the relation between local \(\mathbb{P}\)-shtukas and Galois representations which is given by the associated Tate module. The (dual) Tate functor and the rational (dual) Tate functor are defined. Under certain conditions (dual) Tate functors are equivalences between category of étale local shtukas over \(S\) to the category of finite free \(\mathbb{F}[[z]]\)-modules equipped with continuous action of the algebraic fundamental group of \(S\) at its geometric point. Section 4 works out unbounded Rapoport-Zink spaces for local \(\mathbb{P}\)-shtukas. Boundedness conditions for local \(\mathbb{P}\)-shtukas and bounded local \(\mathbb{P}\)-shtukas also introduced and investigated. Representability of the bounded Rapoport-Zink functor is proved. The results are too technical to be stated here in details. They are applied later to uniformize the moduli stack of global \(\mathfrak{G}\)-shtukas. The authors conclude this impressive work by proving the rigidity of quasi-isogenies for global \(\mathfrak{G}\)-shtukas and by proving the analog of the Serre-Tate theorem.

Reviewer: Nikolaj M. Glazunov (Kyïv)

##### MSC:

14L05 | Formal groups, \(p\)-divisible groups |

11G18 | Arithmetic aspects of modular and Shimura varieties |

14G35 | Modular and Shimura varieties |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |