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## Preprints

Classics revisited: Éléments de géométrie algébrique

April 2018, pdf, submitted to Jahresbericht der DMV, 50 p.

About 50 years ago, Éléments de Géométrie Algébrique (EGA) by
A. Grothendieck and J. Dieudonné appeared, an encyclopedic work on the
foundations of Grothendieck's algebraic geometry. We sketch some of the most
important concepts developed there, comparing it to the classical language, and
mention a few results in algebraic and arithmetic geometry which have since
been proved using the new framework.

Stratifications of affine Deligne-Lusztig varieties

January 2018, arxiv:1802.02225, 21 p.

Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in
the context of affine flag varieties and affine Grassmannians. They are closely
related to moduli spaces of $p$-divisible groups in positive characteristic,
and thus to arithmetic properties of Shimura varieties.
We compare stratifications of affine Deligne-Lusztig varieties attached to
a basic element $b$. In particular, we show that the stratification defined by
Chen and Viehmann using the relative position to elements of the group $\mathbb J_b$,
the $\sigma$-centralizer of $b$, coincides with the Bruhat-Tits stratification
in all cases of Coxeter type, as defined by X. He and the author.

Erratum: $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties

We correct two small mistakes in [G., He, Nie, $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties]. All the main results remain unaffected.

Fully Hodge-Newton decomposable Shimura varieties

with X. He, S. Nie,
arXiv:1610.05381

The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties.

We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms.

Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of \cite{HR}) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions.

While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups.

We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms.

Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of \cite{HR}) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions.

While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups.

## Refereed publications

Basic loci in Shimura varieties of Coxeter type

This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo p of Shimura varieties. Motivated by work of Vollaard-Wedhorn and Rapoport-Terstiege-Wilson, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.

Erratum to: Basic loci in Shimura varieties of Coxeter type

We correct a few mistakes in [G., He, Basic loci in Shimura varieties of Coxeter type]. The main results remain unaffected.

$P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in Görtz et al. (2010). The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.

The supersingular locus in Siegel modular varieties with Iwahori level structure

We study moduli spaces of abelian varieties in positive
characteristic, more specifically the moduli space of
principally polarized abelian varieties on the one hand, and
the analogous space with Iwahori type level structure, on the
other hand. We investigate the Ekedahl-Oort stratification on
the former, the Kottwitz-Rapoport stratification on the latter,
and their relationship. In this way, we obtain structural
results about the supersingular locus in the case of Iwahori
level structure, for instance a formula for its dimension in
case

*g*is even.
Ekedahl-Oort strata and Kottwitz-Rapoport strata

with M. Hoeve, J. Algebra

**351**(2012), 160-174, doi: 10.1016/j.jalgebra.2011.10.039 , arXiv:math/0808.2537
We study the moduli space A_g of g-dimensional principally
polarized abelian varieties in positive characteristic, and its variant
A_I with Iwahori level structure. Both supersingular
Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are
isomorphic to disjoint unions of Deligne-Lusztig varieties (see
[Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these
isomorphisms. We also give an explicit description of Kottwitz-Rapoport
strata contained in the supersingular locus in the general parahoric case.
Finally, we show that every Ekedahl-Oort stratum is isomorphic to a
parahoric Kottwitz-Rapoport stratum.

Dimensions of affine Deligne-Lusztig varieties in affine flag varieties

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig
varieties in the context of an affine root system. We prove a
conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz,
Reuman, and the first named author, about the question which affine
Deligne-Lusztig varieties (for a split group and a basic
$\sigma$-conjugacy class) in the Iwahori case are non-empty. If the
underlying algebraic group is a classical group and the chosen basic
$\sigma$-conjugacy class is the class of $b=1$, we also prove the
dimension formula predicted in op. cit. in almost all cases.

Affine Deligne-Lusztig varieties in affine flag varieties

with T. Haines, R. Kottwitz, D. Reuman,
Compositio Math. 146 (2010), 1339-1382,
arXiv:math/0805.0045v2

This paper studies affine Deligne-Lusztig varieties in the affine
flag manifold of a split group. Among other things, it proves
emptiness for certain of these varieties, relates some of them to
those for Levi subgroups, extends previous conjectures concerning
their dimensions, and generalizes the superset method.

Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties

We investigate Siegel modular varieties in positive characteristic with
Iwahori level structure. On these spaces, we have the Newton
stratification, and the Kottwitz-Rapoport stratification; one would like
to understand how these stratifications are related to each other. We
give a simple description of (conjecturally all) KR strata which are
entirely contained in the supersingular locus as disjoint unions of
Deligne-Lusztig varieties. We also give an explicit numerical description
of the KR stratification in terms of abelian varieties.

On the connectedness of Deligne-Lusztig varieties

We give a criterion which determines when a union of
one-dimensional Deligne-Lusztig varieties has a connected
closure. We also obtain a new, short proof of the connectedness
criterion for Deligne-Lusztig varieties due to Lusztig.

Alcove walks and nearby cycles on affine flag manifolds

Using Ram's theory of alcove walks, we give a proof of the Bernstein
presentation of the affine Hecke algebra. The method works also in the case of
unequal parameters. We also discuss how these results help in studying sheaves
of nearby cycles on affine flag manifolds.

The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties.

We study the Jordan-Hoelder series for nearby cycles on certain Shimura
varieties and Rapoport-Zink local models, and on finite-dimensional pieces of
Beilinson's deformation of the affine Grassmannian to the affine flag variety
(and their p-adic analogues). We give a formula for the multiplicities of
irreducible constituents in terms of certain cohomology groups, and we also
provide an algorithm to compute multiplicities, in terms of the affine Hecke
algebra.

Arithmetic intersection numbers

We define the arithmetic intersection number of three modular divisors and
interpret it from the point of view of algebraic stacks. A criterion is
given when the intersection of three modular divisors is finite.
Furthermore,
the final result about the arithmetic intersection numbers, as given by
Gross and
Keating, is stated and the strategy of its proof, carried out in the
subsequent
chapters, is explained.

A sum of representation numbers

This article contains the proof of a formula stated in the paper by Gross and
Keating on intersections of modular correspondences, for a certain sum of
representation densities.

Dimensions of some affine Deligne-Lusztig varieties

with T. Haines, R. Kottwitz, D. Reuman,
Ann. sci. de l'ENS. 4e serie, t.

**39**(2006), 467-511 , math.AG/0504443.
This paper concerns the dimensions of certain affine Deligne-Lusztig varieties,
both in the affine Grassmannian and in the affine flag manifold. Rapoport
conjectured a formula for the dimensions of the varieties X_mu(b) in the affine
Grassmannian. We prove his conjecture for b in the split torus; we find that
these varieties are equidimensional; and we reduce the general conjecture to
the case of superbasic b. In the affine flag manifold, we prove a formula that
reduces the dimension question for X_x(b) with b in the split torus to
computations of dimensions of intersections of Iwahori orbits with orbits of
the unipotent radical. Calculations using this formula allow us to verify a
conjecture of Reuman in many new cases, and to make progress toward a
generalization of his conjecture.

Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds.

We study certain nearby cycles sheaves on an affine flag manifold which arise
naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to
the affine Grassmannian. We study the multiplicity functions we introduced in
an earlier paper, which encode the data of the Jordan-Hoelder series. We prove
the multiplicity functions are polynomials in q, and we give a sharp bound for
their degrees. Our results apply as well to the nearby cycles in the p-adic
deformation of Laumon-Haines-Ngo, and also to Wakimoto sheaves.

Topological flatness of local models in the ramified case.

Local models are schemes defined in terms of linear algebra which can be used
to study the local structure of integral models of certain Shimura varieties,
with parahoric level structure. We investigate the local models for groups of
the form Res

_{F/Qp}GL_{n}and Res_{F/Qp}GSp_{2n}, where*F*/**Q**_{p}is a totally ramified extension, as defined by Pappas and Rapoport, and show that they are topologically flat. In the linear case, flatness can be deduced from this.
Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for GL_4 and GL_5.

Consider a Shimura variety defined over some number field, and assume we have a
model over the ring of integers at some prime of bad reduction. It is then
interesting to know the alternating trace of Frobenius on the invariants under
the inertia group of the sheaf of nearby cycles, since these traces are related
to the local factor of the Hasse-Weil zeta function. In this article we
compute the semi-simple alternating trace of Frobenius for Shimura varieties
associated to the groups GU(2,2) and GU(3,2), with level structure of Iwahori
type, by investigating the equations of the local model defined by Rapoport and
Zink and by performing explicit blowing ups.

On the flatness of local models for the symplectic group

We investigate the bad reduction of certain Shimura varieties (associated to
the symplectic group). More precisely, we look at a model of the Shimura
variety at a prime p, with parahoric level structure at p. We show that this
model is flat, as conjectured by Rapoport and Zink, and that its special fibre
is reduced.

On the flatness of models of certain Shimura varieties of PEL type

Consider a PEL-Shimura variety associated to a unitary group that splits over
an unramified extension of Q_p. Rapoport and Zink have defined a model of the
Shimura variety over the ring of integers of the completion of the reflex field
at a place lying over p, with parahoric level structures at p. We show that
this model is flat, as conjectured by Rapoport and Zink, and that its special
fibre is reduced.

Coherent modules and their descent on relative rigid spaces.

with S. Bosch
J. Reine Angew. Math.

**495**(1998), 119-134
We prove that faithfully flat descent holds for coherent modules on (classical)
rigid spaces, also over non-noetherian bases.

## Books

Algebraic geometry I. Schemes, with Examples and Exercises

with T. Wedhorn, Vieweg (2010), 615 p.

## Non-refereed publications/contributions to proceedings volumes

Book review: James Arthur,

*The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups.*AMS Colloquium Publ. 61 (2013)
in: Jahresbericht der DMV

**116**, Nr. 2 (2014) 123-128.
Abel-Preis für Pierre Deligne

Mitteilungen der DMV

**21.3**(2013), 151-155
Eine Fields-Medaille für Bao Châu Ngô

Mitteilungen der DMV

**19.4**(2011), 198-203
Affine Springer fibers and affine Deligne-Lusztig varieties

in: A. Schmitt (ed.), Proceedings of Affine Flag Manifolds and Principal
Bundles (Berlin 2008), Trends in Mathematics, Birkhäuser (2010), pdf

Reduction of Shimura varieties and Deligne-Lusztig varieties

in Math. Forschungsinst. Oberwolfach Report 2010, no. 19, 1121-1124,
pdf

Matrixgleichungen und Familien abelscher Varietäten in positiver Charakteristik

Mitteilungen der DMV

**17.1**(2009) pdf
The supersingular locus in Siegel modular varieties, and Deligne-Lusztig varieties

in Math. Forschungsinst. Oberwolfach Report 2008, no. 5, 283-286,
pdf.

Affine Deligne-Lusztig varieties

in Math. Forschungsinst. Oberwolfach Report 2008, no. 3, 136-139
pdf

Affine Deligne-Lusztig varieties

in: Math. Forschungsinst. Oberwolfach, Report 2007, no. 30, 1785-1787

Lokale Modelle von Shimura-Varietäten und ihre Garben der verschwindenden Zykel

in: Y. Tschinkel, Seminars 2003/04, Math. Inst. der Univ. Göttingen.

Reviews of these papers in MathSciNet.