
Normed Fiber Functors
Abstract: By GoldmanIwahori, the BruhatTits building of the general linear group GLn over a local field l can be described as the set of nonarchimedean norms on l^n. Via a Tannakian formalism, we generalize this picture to a description of the BruhatTits building of an arbitrary 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 modulitheoretic description of the BruhatTits 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.

Tautological rings of Shimura varieties and cycle classes of EkedahlOort strata (with Torsten Wedhorn)
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 EkedahlOort strata as a vector space. We compute these cycle classes. As applications we get the triviality of ladic 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 HirzebruchMumford proportionality for Shimura varieties of Hodge type.
Publications

Geometric stabilisation via padic integration (with Michael Groechenig and Dimitri Wyss)
J. Amer. Math. Soc. 33 (2020), no. 3, 807873.
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 quasisplit 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 padic integration on coarse moduli spaces of DeligneMumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of GHiggs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasisplit case.

Mirror symmetry for moduli spaces of Higgs bundles via padic integration (with Michael Groechenig and Dimitri Wyss)
Invent. Math. 221 (2020), no. 2, 505596.
Abstract: We prove the Topological Mirror Symmetry Conjecture by HauselThaddeus 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 padic 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 MozgovoySchiffman in the coprime case.

pkernels occurring in isogeny classes of pdivisible groups
Bulletin of the LMS, Volume 49, Issue 2, 2017, pp. 185367.
Abstract: We give a criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which pkernels occur in an isogeny class of pdivisible groups over an algebraically closed field of positive characteristic. As an application we obtain a criterion for the nonemptiness of certain affine DeligneLusztig varieties associated to the general linear group.

MordellLang in positive characteristic
Rendiconti del Seminario Matematico della Università di Padova, Volume 134, 2015, pp. 93131.
Abstract: We give a new proof of the MordellLang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full MordellLang conjecture in positive characteristic.

FZips with additional structure (with Richard Pink and Torsten Wedhorn)
Pacific Journal of Mathematics, Vol. 274 (2015), No. 1, 183236.
Abstract: An Fzip over a scheme S over a finite field is a certain object of semilinear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a \Frob_qtwisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "Fzips with a Gstructure", 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 Fzips over S. Locally any such functor has a type χ, which is a cocharacter of G. The other incarnation is a certain Gtorsor analogue of the notion of Fzips. 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 Fzips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper DeligneMumford stacks) and to truncated BarsottiTate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the EkedahlOort 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. 87130.
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 fpqclocally on the base scheme.

Presentations for quaternionic Sunit groups (with Ted Chinburg, Holley Friedlander, Sean Howe, Michiel Kosters, Bhairav Singh, Matthew Stover and Ying Zhang)
Experimental Mathematics (2015), 24:2, 175182.
Abstract: The purpose of this paper is to give presentations for projective Sunit 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 Sarithmetic lattice in a semisimple Lie group with S large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of BruhatTits 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) 253300
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×Qϕ(πP(p))=πQ(q)} on G given by ((p,q),g)=pgq^{−1}. We define certain smooth Einvariant 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 Eaction on G. We also generalize all results to nonconnected 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 Gvariety 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 Fzips. In these cases the Einvariant subvarieties correspond to the socalled "Gstable pieces" of Z defined by Lusztig (resp. the Gorbits of Z′).