[Paper Review] Local P-shtukas and their relation to global G-shtukas

Esmail M. Arasteh Rad, Urs Hartl|arXiv (Cornell University)|Feb 25, 2013

Algebraic Geometry and Number Theory39 references22 citations

TL;DR

This paper establishes the foundational link between global G-shtukas—function field analogs of Shimura varieties—and local P-shtukas, which generalize p-divisible groups with additional structure. It proves a Serre–Tate-type theorem showing equivalence between deformations of global G-shtukas and their associated local P-shtukas, and constructs Rapoport–Zink spaces for bounded local P-shtukas as formal schemes locally formally of finite type, enabling uniformization of global G-shtuka moduli stacks in subsequent work.

ABSTRACT

This is the first in a sequence of two articles investigating moduli stacks of global G-shtukas, which are function field analogs for Shimura varieties. Here G is a flat affine group scheme of finite type over a smooth projective curve, and global G-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. Our moduli stacks generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the present article we explain the relation between global G-shtukas and local P-shtukas, which are the function field analogs of p-divisible groups with additional structure. We prove the analog of a theorem of Serre and Tate stating the equivalence between the deformations of a global G-shtuka and its associated local P-shtukas. We also investigate local P-shtukas alone and explain their relation with Galois representations through their Tate modules. And if P is a smooth affine group scheme with connected reductive generic fiber we prove the existence of Rapoport--Zink spaces for bounded local P-shtukas as formal schemes locally formally of finite type. In the sequel to this article we use these Rapoport--Zink spaces to uniformize the moduli stacks of global G-shtukas.

Motivation & Objective

• To establish a precise correspondence between global G-shtukas and local P-shtukas, generalizing the role of p-divisible groups in p-adic Hodge theory.
• To prove a function field analog of the Serre–Tate theorem, relating infinitesimal deformations of global G-shtukas to those of their associated local P-shtukas.
• To construct Rapoport–Zink spaces for bounded local P-shtukas as formal schemes locally formally of finite type, extending classical p-adic uniformization.
• To provide a framework for uniformizing moduli stacks of global G-shtukas using local P-shtuka deformation spaces, as a key step toward realizing the Langlands correspondence over function fields.

Proposed method

• Introduces local P-shtukas as tuples (L+, ˆτ) involving L+Pν-torsors and Frobenius-twisted isomorphisms ˆτ: ˆσ∗L → L, generalizing Drinfeld’s shtukas.
• Develops a global-local functor that associates to a global G-shtuka over a scheme in NilpAν a tuple of local Pνi-shtukas at characteristic places νi.
• Applies a generalized Beauville–Laszlo gluing lemma to construct the global-local correspondence over formal completions.
• Proves representability of the unbounded Rapoport–Zink functor by an ind-scheme over Spf k[[ζ]], and establishes boundedness conditions via Hodge polygon control.
• Uses axiomatic bounds on the relative position of ˆσ∗L+ and L+ under ˆτ to define bounded local P-shtukas.
• Applies techniques inspired by Rapoport–Zink and Hartl to prove that the bounded Rapoport–Zink functor is representable by a formal scheme locally formally of finite type over Spf k[[ζ]].

Experimental results

Research questions

• RQ1 How do global G-shtukas relate to their associated local P-shtukas at characteristic places?
• RQ2 Is there a Serre–Tate-type equivalence between the deformation theory of global G-shtukas and that of their local P-shtuka counterparts?
• RQ3 Can Rapoport–Zink spaces for bounded local P-shtukas be constructed as formal schemes locally formally of finite type?
• RQ4 What is the role of Tate modules in connecting local P-shtukas to Galois representations?
• RQ5 How can the moduli stack of global G-shtukas be uniformized using local P-shtuka deformation spaces?

Key findings

• The deformation functors of global G-shtukas and their associated n-tuples of local Pνi-shtukas are equivalent, generalizing the classical Serre–Tate theorem to the function field setting.
• The unbounded Rapoport–Zink functor for local Pν-shtukas is representable by an ind-scheme, specifically isomorphic to FℓPν b×Fν Spf Fν[[ζ]] when the L+Pν-torsor is trivial.
• For smooth Pν with connected reductive generic fiber, the bounded Rapoport–Zink functor is representable by a formal scheme locally formally of finite type over Spf k[[ζ]], as proven in Theorem 4.18.
• The construction of Rapoport–Zink spaces for bounded local P-shtukas provides a key technical tool for the uniformization of moduli stacks of global G-shtukas in the sequel.
• Tate modules of local P-shtukas give rise to Galois representations, establishing a link between the geometry of local P-shtukas and the Langlands program.
• The results lay the foundation for realizing the Langlands correspondence for function fields via the cohomology of moduli stacks of global G-shtukas.