arXiv subjects
Publications and source records attributed to Urs Hartl.
A finite dimensional real vector space carrying an action of a real reductive group possesses a moment map and a stratification as defined by Kirwan and Ness. In this article we investigate the properties of the moment map, the Kirwan-Ness stratification, and Lauret's application from a more functorial, algebraic point of view.
In this article we formulate and prove the analogue of the Langlands-Rapoport conjecture for the moduli stacks of global $G$-shtukas. Here $G$ is a parahoric Bruhat-Tits group scheme over a smooth projective curve $C$ over a finite field $\mathbb{F}_q$.
We show that the moduli spaces of bounded global $\mathcal{G}$-Shtukas with pairwise colliding legs admit $p$-adic uniformization isomorphisms by Rapoport-Zink spaces. Here $\mathcal{G}$ is a smooth affine group scheme with connected fibers and reductive generic fiber, i.e. we do not assume it to be parahoric, or even hyperspecial. Moreover, we deduce the Langlands-Rapoport Conjecture over function fields in the case of colliding legs using our uniformization theorem.
Moduli spaces of bounded local $G$-shtukas are a group-theoretic generalization of the function field analog of Rapoport and Zink's moduli spaces of $p$-divisible groups. In this article we generalize some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local $G$-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fiber of the moduli space which is equipped with a Hecke action and an action of a suitable automorphism group. Finally we consider the $\ell$-adic cohomology of these towers.
Colmez conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez's theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher dimensional generalizations, so-called $A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $A$-motive at all finite places in terms of Artin $L$-series. The latter is achieved by investigating the local shtukas associated with the $A$-motive.
In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and is more relevant for applications to the theory of $G$-Shtukas, such as formulating the analog of the Langlands-Rapoport conjecture over function fields. We further develop the analogy with the category of motives over $L$ with coefficients in $\mathbb{Q}$ for which the existence of the expected fiber functors depends on famous unproven conjectures.
In this article we construct the arithmetic Satake compactification of the Drinfeld moduli schemes of arbitrary rank over the ring of integers of any global function field away from the level structure, and show that the universal family extends uniquely to a generalized Drinfeld module over the compactification. Using these and functorial properties, we define algebraic Drinfeld modular forms over more general bases and the action of the (prime-to-residue characteristic and level) Hecke algebra. The construction also furnishes many algebraic Drinfeld modular forms obtained from the coefficients of the universal family which are also Hecke eigenforms. Among them we obtain generalized Hasse invariants which are already defined on the arithmetic Satake compactification and not only its special fiber. We use these generalized Hasse invariants to study the geometry of the special fiber. We conjecture that our Satake compactification is Cohen-Macaulay. If this is the case, we establish the Jacquet-Langlands correspondence (mod $v$) between Hecke eigensystems of rank $r$ Drinfeld modular forms and those of algebraic modular forms (in the sense of Gross) attached to a compact inner form of $GL_r$.
We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.
We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and these cohomology realizations arise from Drinfeld modules and Anderson's $t$-motives. As an application we construct equi-characteristic crystalline deformation rings, establish their rigid-analytic smoothness and compute their dimension.
In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian $t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable $t$-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analog of the Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished work of Pink and the first author, the second author proved in her Diploma thesis that the Hodge-Pink group equals the motivic Galois group of the $t$-motive as defined by Papanikolas and Taelman. This yields a precise analog of the famous Hodge Conjecture, which is an outstanding open problem for varieties over the complex numbers. In this report we explain Pink's results on Hodge structures and the proof of the function field analog of the Hodge conjecture. The theory of $t$-motives has a variant in the theory of dual $t$-motives. We clarify the relation between $t$-motives, dual $t$-motives and $t$-modules. We also construct cohomology realizations of abelian $t$-modules and (dual) $t$-motives and comparison isomorphisms between them generalizing Gekeler's de Rham isomorphism for Drinfeld modules.
Let $K$ be a finite field extension of $Q_p$ and let $G_K$ be its absolute Galois group. We construct the universal family of filtered $(ϕ,N)$-modules, or (more generally) the universal family of $(ϕ,N)$-modules with a Hodge-Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semi-stable $G_K$-representations in $Q_p$-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over $Q_p$ in the sense of Huber. This has conjectural applications to the $p$-adic local Langlands program.
This is the second in a sequence of articles, in which we explore moduli stacks of global G-shtukas, the function field analogs for Shimura varieties. Here G is a flat affine group scheme of finite type over a smooth projective curve C over a finite field. Global G-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global G-shtukas are algebraic Deligne-Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the first article we explained the relation between global G-shtukas and local P-shtukas, which are the function field analogs of p-divisible groups. Here P is the base change of G to the complete local ring at a point of C. When P is smooth with connected reductive generic fiber we proved the existence of Rapoport-Zink spaces for local P-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global G-shtukas for smooth G with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands-Rapoport conjecture for our moduli stacks.
We develop the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings's and Abrashkin's theory of strict modules, which we review to some extent.
Using the Tannakian formalism, we formulate conjectural analogs of Chebotar\"ev's Density Theorem for $F$-isocrystals over a smooth geometrically irreducible variety defined over a finite field. We prove these analogs for several large classes, including (a) constant $F$-isocrystals, (b) direct sums of isoclinic convergent $F$-isocrystals, (c) semi-simple overconvergent $F$-isocrystals, and (d) semi-simple convergent $F$-isocrystals which have an overconvergent extension. Case (a) is a generalization of the Mordell-Lang Conjecture for tori and enters in the proofs of (b) and (c). For (b) we use the classical Chebotar\"ev Density Theorem, and point counting techniques in $p$-adic Lie groups building on a result of Oesterl\'e. For (c) we give two proofs. One of them uses deep input on the Langlands correspondence by Abe and Lafforgue, and the theory of Frobenius weights of Kedlaya, Abe and Caro. Building on this we formulate and prove the $p$-adic analog of Deligne's Equidistribution Theorem. Then (c) follows by applying real algebraic geometry to maximal compact subgroups in complex algebraic groups, measure theory, and a convergence result on complex hypersurfaces. For (d) we develop the theory of maximal quasi-tori (generalizing maximal tori in non-connected linear algebraic groups) and use D'Addezio's result on Crew's parabolicity conjecture to reduce to (b). These arguments also yield a second proof of (c). Besides of the deep inputs mentioned above and some Tannakian arguments, our main technique is the theory of linear algebraic groups. We include a comparison with the recent article of Cadoret and Tamagawa on the same topic.
As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the associated t-motive via the finite shtuka correspondence of Drinfeld and Abrashkin. The inductive limits of torsion submodules are the function field analogs of p-divisible groups. These limits correspond to the local shtukas attached to the t-motives associated with the abelian t-modules. In this sense the theory of abelian t-modules is captured by the theory of t-motives.
For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck's and de Jong's criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group.
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.
In their book Rapoport and Zink constructed rigid analytic period spaces $F^{wa}$ for Fontaine's filtered isocrystals, and period morphisms from PEL moduli spaces of $p$-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism $F^a \to F^{wa}$ of rigid analytic spaces and of a universal local system of $Q_p$-vector spaces on $F^a$. For Hodge-Tate weights $n-1$ and $n$ we construct in this article an intrinsic Berkovich open subspace $F^0$ of $F^{wa}$ and the universal local system on $F^0$. We conjecture that the rigid-analytic space associated with $F^0$ is the maximal possible $F^a$, and that $F^0$ is connected. We give evidence for these conjectures and we show that for those period spaces possessing PEL period morphisms, $F^0$ equals the image of the period morphism. Then our local system is the rational Tate module of the universal $p$-divisible group and enjoys additional functoriality properties. We show that only in exceptional cases $F^0$ equals all of $F^{wa}$ and when the Shimura group is $GL_n$ we determine all these cases.