## PD Dr. Alexander Ivanov

Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn

Room: 1.004
Telefon: +49(0)228-733791
E-Mail: ivanov"at"math.uni-bonn.de
Group: Arithmetische Algebraische Geometrie

Recent

• April 16, 2021: Inaugural lecture at the University of Bonn. (In German, topic: "The sphere packing problem") The slides are available here.
• In February 2021 I was habilitated at the University of Bonn.
• On January 7, 2021 I gave a talk in the RAMpAGe Seminar. The talk (notes and video) is available at this link.

Teaching

In the summer term (2021) I will give a lecture on etale cohomology. The lecture will take place Mondays 8-10, via zoom. See here for more information.
Click here to see my teaching in the previous semesters.

Research

I am interested in arithmetic geometry, representation theory and number theory. My current projects are related to
• p-adic analogues of Deligne-Lusztig theory
• representations of p-adic reductive groups and local Langlands correspondences
• modifications of vector bundles on the Fargues--Fontaine curve, related perfectoid phenomena and Rapoport--Zink spaces at infinite level
I also worked on (and am still interested in) the following topics:
• restricted ramification in number fields, Hasse principles, Grunwand--Wang-style results
• densities of primes in number fields, generalizations of Dirichlet density
• anabelian geomerty, especially in the context of rings of integers in number fields

In the winter term 2018/19 I was deputy W1-professor at the Goethe-Universität Frankfurt.
Link to my homepage there.

Preprints

1. Orthogonality relations for deep level Deligne-Lusztig schemes of Coxeter type (with Olivier Dudas)
preprint 2020, submitted.
[ arXiv ]
Abstract Orthogonality relations in the classical Deligne-Luszig theory compute the inner product between two Deligne-Lusztig characters as some explicit expression in terms of the Weyl group. They form an important cornerstone of the whole theory. For deep level Deligne-Lusztig varieties a similar result in full generality is still open. In this article we prove it in the special case of Coxeter varieties, but without any assumption on the involved characters.
2. On ind-representability of $$p$$-adic Deligne-Lusztig spaces
preprint 2020, submitted.
[ arXiv ]
AbstractWe give a new definition of $$p$$-adic Deligne-Lusztig spaces $$X_w(b)$$ using the loop functor. We prove that they are arc-sheaves on perfect schemes over the residue field. We establish some fundamental properties of $$X_w(b)$$ and the natural torsors on them. In particular, we show that $$X_w(b)$$ is ind-representable if $$w$$ has minimal length in its $$\sigma$$-conjugacy class. Along the way we show two general results: first, for a quasi-projective scheme $$X$$ over a local non-archimedean field $$k$$, the loop space $$LX$$ is an arc-sheaf (this uses perfectoid methods). Second, for an unramified reductive group $$G$$ over $$k$$ with a Borel subgroup $$B$$, $$LG \rightarrow L(G/B)$$ is surjective in the $$v$$-topology.
3. On loop Deligne-Lusztig varieties of Coxeter type for inner forms of $${\rm GL}_n$$ (with Charlotte Chan)
preprint 2019, submitted.
[ arXiv ]
AbstractWe study the natural torsor over the $$p$$-adic Deligne-Lusztig space $$X_w(b)$$ attached to the group $${\rm GL}_n$$, Coxeter element $$w$$ and basic $$b$$. We show that it is representable by a scheme and study its $$\ell$$-adic cohomology. Our main result is that the latter realizes many irreducible supercuspidal representations of $${\rm GL}_n(k)$$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $${\rm GL}_n$$. This gives a purely local and geometric way to realize many special cases of the local Langlands and Jacquet--Langlands correspondences.
4. The Drinfeld stratification for $${\rm GL}_n$$ (with Charlotte Chan)
preprint 2020, submitted.
[ arXiv ]
5. Reconstructing decomposition subgroups in arithmetic fundamental groups using regulators
preprint 2014, submitted.
[ arXiv ]

Published (or accepted) articles

1. Cohomological representations of parahoric subgroups (with Charlotte Chan)
Representation Theory 25 (2021), 1-26.
[ arXiv ] [ Journal ]
AbstractGeneralizing Lusztig's work, we give a geometric construction of representations of parahoric subgroups $$P$$ of a reductive group $$G$$ over a local field which splits over an unramified extension. These representations correspond to characters $$\theta$$ of unramified maximal tori and, when the torus is elliptic, are expected give rise to supercuspidal representations of $$G$$. We calculate the character of these $$P$$-representations on a special class of regular semisimple elements of $$G$$. Under a certain regularity condition on $$\theta$$, we prove that the associated $$P$$-representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric.
2. The smooth locus in infinite-level Rapoport-Zink spaces (with Jared Weinstein)
Compositio Mathematica 156 (2020), No. 9, 1846-1872.
[ arXiv ] [ Journal ]
AbstractRapoport-Zink spaces are deformation spaces for $$p$$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $$\mathcal{M}_{\infty}$$ be an infinite-level Rapoport-Zink space of EL type, and let $$\mathcal{M}_{\infty}^\circ$$ be one connected component of its geometric fiber. We show that $$\mathcal{M}_{\infty}^{\circ}$$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $$p$$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $$X(p^\infty)^{\circ}$$ is exactly the locus of elliptic curves $$E$$ with supersingular reduction, such that the formal group of $$E$$ has no extra endomorphisms.
3. Affine Deligne-Lusztig varieties at infinite level (with Charlotte Chan)
Mathematische Annalen (to appear)
[ arXiv ]

4. Ordinary GL2(F)-representations in characteristic two via affine Deligne-Lusztig constructions
Mathematical Research Letters 27 (2020), No. 1, 141-187.
[ arXiv ] [ Journal ]

5. Ramified automorphic induction and zero-dimensional affine Deligne-Lusztig varieties
Mathematische Zeitschrift 288 (2018), 439-490.
[ arXiv ] [ Journal ]

6. Densities of primes and realization of local extensions
Transactions Amer. Math. Soc. 371 (2019), 83-103.
[ arXiv ] [ Journal ]

7. On a generalization of the Neukirch-Uchida theorem
Moscow Mathematical Journal 17 (2017), no. 3, 371-383.
[ arXiv ] [ Journal ]

8. Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for GL2
Advances in Mathematics 299 (2016), 640-686.
[ arXiv ] [ Journal ]

9. Stable sets of primes in number fields
Algebra & Number Theory 10 (2016), No. 1, 1-36.
[ arXiv ] [ Journal ]

10. On some anabelian properties of arithmetic curves
Manuscripta Mathematica 144 (2014), No. 3, 545-564.
[ arXiv ] [ Journal ]

11. Cohomology of affine Deligne-Lusztig varieties for GL2
Journal of Algebra 383 (2013), 42-62.
[ arXiv ] [ Journal ]

Here are all my articles on arXiv.

Submitted theses (Abschlussarbeiten)

1. p-adic Deligne--Lusztig theory
Habilitation thesis, Bonn, 2020.
The introduction may be found here. The full version might be available upon request.

2. Arithmetic and anabelian theorems for stable sets of primes in number fields
Ph.D. thesis, Heidelberg, 2013

3. The cohomology of affine Deligne Lusztig varieties in the affine flag manifold of GL2
Diploma thesis, Bonn, 2009

Non-refereed reports

1. The smooth locus in infinite level Rapoport--Zink spaces
in: Math. Forschungsinst. Oberwolfach, Oberwolfach, Report No. 2/2019, 84-87.
2. Generalized densities of primes and realization of local extensions
in: Math. Forschungsinst. Oberwolfach, Report No. 25/2018, 1538-1540.
3. Affine Deligne-Lusztig varieties of higher level and Local Langlands correspondence for GL2
in: Math. Forschungsinst. Oberwolfach, Report No. 39/2015, 547-551.

Notes not intended for journal publication

CV
Here is my CV.