Recent developments on compactifications of stacks of shtukas

arXiv: Number Theory, 2020

This communication is an introduction to the Langlands Program and to ($G$-) shtukas (over algebraic curves) over function fields. Modular curves and Drinfeld (elliptic) modules and shtukas are used in coding theory. From this point of view the communication is concerned with mathematical models and methods of coding theory. For a connected reductive group $ G $ over a global field $ K $, the Langlands correspondence relates automorphic forms on $ G $ and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb Q}}_p) $. In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. In this review article, we first present results on Langlands program and related representation o...

Memoirs of the American Mathematical Society, 2008

Introduction Reminders on abstract algebraic geometry The setting Linear and commutative algebra in a symmetric monoidal model category Geometric stacks Infinitesimal theory Higher Artin stacks (after C. Simpson) Derived algebraic geometry: D −-stacks Complicial algebraic geometry: D-stacks Brave new algebraic geometry: S-stacks Relations with other works Acknowledgments Notations and conventions Part 1. General theory of geometric stacks Chapter 2.4. Brave new algebraic geometry 2.4.1. Two HAG contexts 2.4.2. Elliptic cohomology as a Deligne-Mumford S-stack Appendix A. Classifying spaces of model categories Appendix B. Strictification

Mathematische Zeitschrift, 2006

We study the relationship between the deformation theory of representable 1-morphisms between algebraic stacks and the cotangent complex defined by Laumon and Moret-Bailly. Z • (δ) * (pr 2 * 12 (σ −1))

Mathematische Annalen, 1976

Algebraic and Geometric Topology, 2008

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and theétale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.

International Mathematics Research Notices

Projective duality identifies the moduli spaces $\textbf{B}_n$ and $\textbf{X}(3,n)$ parametrizing linearly general configurations of $n$ points in $\mathbb{P}^2$ and $n$ lines in the dual $\mathbb{P}^2$, respectively. The space $\textbf{X}(3,n)$ admits Kapranov’s Chow quotient compactification $\overline{\textbf{X}}(3,n)$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $\mathbb{P}^2$ with $n$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $\mathbb{P}^2$ with $n$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.

Advances in Mathematics, 2008

Let C be a smooth projective curve of genus g ≥ 2 over a field k. Given a line bundle L on C, let Sympl 2n,L be the moduli stack of vector bundles E of rank 2n on C endowed with a nowhere degenerate symplectic form b : E ⊗ E −→ L up to scalars. We prove that this stack is birational to BGm × A s for some s if deg(E) = n • deg(L) is odd and C admits a rational point P ∈ C(k) as well as a line bundle ξ of degree 0 with ξ ⊗2 ∼ = O C. It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.

manuscripta mathematica, 2016

Oort stratification and the Kottwitz-Rapoport stratification. The classic work on the first two kinds of stratifications concerns the Siegel case with hyperspecial level structure which was studied by F. Oort and others. There is also a lot of work on other Shimura varieties which are familiar moduli spaces of abelian varieties, comp. the references in . Here we are concerned with the group-theoretic versions of these stratifications . To the stratifications above we add here the Ekedahl-Kottwitz-Oort-Rapoport stratification which interpolates between the Kottwitz-Rapoport stratification in the case of an Iwahori level structure and the Ekedahl-Oort stratification of Viehmann in the hyperspecial case. Concerning these stratifications, one can ask many interesting questions (when do they have the strong stratification property?, when are the strata equi-dimensional and what is their dimension? what are local and global properties of the individual strata-are they smooth, are they (quasi-)affine?, and so on). There is a large body of literature on these topics, but here we are only concerned with the question of which strata are non-empty and of the relation between these various stratifications. The background of these questions is addressed in the survey paper by Haines [10] and the survey paper of the second author , where early work on these problems is described. As far as the non-emptiness of Newton strata in the group-theoretic setting (in their natural index set) is concerned, the goal is to prove the conjecture of the second author [41, Conj. 7.1], comp. also Fargues [6, p. 55] and [10, Conj. 12.2]. This question is considered by Viehmann/Wedhorn in the PEL-case, for hyperspecial level structures. Recent work of D.-U. Lee [34], M. addresses this problem for Shimura varieties of PEL-type, and even of Hodge type, when the underlying group is unramified at p. Kisin proves the non-emptiness of the basic stratum even when the unramifiedness assumption is dropped. The method used in all these works is based on the papers [33] and . There is also work by Kret [31, for Shimura varieties of PEL-type which, besides the papers , also uses the Arthur trace formula.

Nuclear Physics B, 1997

By identifying the moduli space of coupling constants in the SYM description of toroidal compactifications of M(atrix)-theory, we construct the M(atrix) description of the moduli spaces of type IIA string theory compactified on Tn. Addition of theta terms to the M(atrix) SYM produces the shift symmetries necessary to recover the correct global structure of the moduli spaces. Up to n

In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general characterization of heterotic strings on stacks, we specialize to gerbes, and consider three different classes of `building blocks' of gerbe compactifications. We argue that heterotic string compactifications on one class is equivalent to compactification of the same heterotic string on a disjoint union of spaces, compactification on another class is dual to compactifications of other heterotic strings on spaces, and co...