Paul Ziegler

Preprints

• Twisted points of quotient stacks, integration and BPS-invariants (with Michael Groechenig and Dimitri Wyss)
  
  Abstract: We study p-adic manifolds associated with twisted points of quotient stacks X=[U/G] and their quotient spaces π:X→X. We prove several structural results about the fibres of π and derive in particular a formula expressing p-adic integrals on X in terms of the cyclotomic inertia stack of X, generalizing the orbifold formula for Deligne-Mumford stacks. We then apply our formalism to moduli stacks of 1-dimensional sheaves on del Pezzo surfaces and show that their refined BPS-invariants are computed locally on the coarse moduli space by a p-adic integral. As a consequence we recover the χ-independence of these invariants previously proven by Maulik-Shen. Along the way we derive a new formula for the plethystic logarithm on the λ-ring of functions on k-linear stacks, which might be of independent interest.

• Normed Fiber Functors
  
  Abstract: By Goldman-Iwahori, the Bruhat-Tits building of the general linear group GLn over a local field l can be described as the set of non-archimedean norms on l^n. Via a Tannakian formalism, we generalize this picture to a description of the Bruhat-Tits building of an unramified reductive group G over l as the set of norms on the standard fiber functor of a special parahoric integral model of G. We also give a moduli-theoretic description of the parahoric group scheme associated to a point of the building as the group scheme of tensor automorphisms of the lattice chains defined by the corresponding norm.

Publications

• Filtered fiber functors over a general base
  To appear in: Transformation Groups
  
  Abstract: We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.

• Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort strata (with Torsten Wedhorn)
  Algebra Number Theory, 17(4):923–980, 2023.
  
  Abstract: We define the tautological ring as the subring of the Chow ring of a Shimura variety generated by all Chern classes of all automorphic bundles. We explain its structure for the special fiber of a good reduction of a Shimura variety of Hodge type and show that it is generated by the cycle classes of the Ekedahl-Oort strata as a vector space. We compute these cycle classes. As applications we get the triviality of l-adic Chern classes of flat automorphic bundles in characterstic 0, an isomorphism of the tautological ring of smooth toroidal compactifications in positive characteristic with the rational cohomology ring of the compact dual of the hermitian domain given by the Shimura datum, and a new proof of Hirzebruch-Mumford proportionality for Shimura varieties of Hodge type.

• Geometric stabilisation via p-adic integration (with Michael Groechenig and Dimitri Wyss)
  J. Amer. Math. Soc. 33 (2020), no. 3, 807-873.
  
  Abstract: In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme G to the cohomology of Hitchin fibres for the endoscopy groups H_κ. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for p-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of G-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.


• Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration (with Michael Groechenig and Dimitri Wyss)
  Invent. Math. 221 (2020), no. 2, 505-596.
  
  Abstract: We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SL_n and PGL_n. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for d coprime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffman in the coprime case.


• p-kernels occurring in isogeny classes of p-divisible groups
  Bulletin of the LMS, Volume 49, Issue 2, 2017, pp. 185-367.
  
  Abstract: We give a criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which p-kernels occur in an isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. As an application we obtain a criterion for the non-emptiness of certain affine Deligne-Lusztig varieties associated to the general linear group.


• Mordell-Lang in positive characteristic
  Rendiconti del Seminario Matematico della Università di Padova, Volume 134, 2015, pp. 93-131.
  
  Abstract: We give a new proof of the Mordell-Lang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full Mordell-Lang conjecture in positive characteristic.


• F-Zips with additional structure (with Richard Pink and Torsten Wedhorn)
  Pacific Journal of Mathematics, Vol. 274 (2015), No. 1, 183-236.
  
  Abstract: An F-zip over a scheme S over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a \Frob_q-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "F-zips with a G-structure", for an arbitrary reductive linear algebraic group G. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of G to the category of F-zips over S. Locally any such functor has a type χ, which is a cocharacter of G. The other incarnation is a certain G-torsor analogue of the notion of F-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form [E_{G,χ}\ G_k] that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over k. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary F-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.


• Graded and Filtered Fiber Functors on Tannakian Categories
  Journal of the Institute of Mathematics of Jussieu, volume 14 (2015), issue 01, pp. 87-130.
  
  Abstract: We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme.


• Presentations for quaternionic S-unit groups (with Ted Chinburg, Holley Friedlander, Sean Howe, Michiel Kosters, Bhairav Singh, Matthew Stover and Ying Zhang)
  Experimental Mathematics (2015), 24:2, 175-182.
  
  Abstract: The purpose of this paper is to give presentations for projective S-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field Q. To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when S contains at least two odd primes. In the appendix, we provide code that allows the reader to compute presentations for an arbitrary finite set S.


• Algebraic Zip Data (with Richard Pink and Torsten Wedhorn)
  Documenta Math. 16 (2011) 253-300
  
  Abstract: An algebraic zip datum is a tuple Z = (G,P,Q,\phi) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny ϕ:P/RuP→Q/RuQ. We study the action of the group E:={(p,q)∈P×Q|ϕ(πP(p))=πQ(q)} on G given by ((p,q),g)=pgq^{−1}. We define certain smooth E-invariant subvarieties of G, show that they define a stratification of G. We determine their dimensions and their closures and give a description of the stabilizers of the E-action on G. We also generalize all results to non-connected groups. We show that for special choices of Z the algebraic quotient stack [E\G] is isomorphic to [G\Z] or to [G\Z′], where Z is a G-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of G and where Z′ has been defined by Moonen and the second author in their classification of F-zips. In these cases the E-invariant subvarieties correspond to the so-called "G-stable pieces" of Z defined by Lusztig (resp. the G-orbits of Z′).