## 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

I am a Heisenberg fellow of the DFG, working at the University of Bonn. I am interested in arithmetic geometry, number theory and representation theory. Here are some keywords describing my current research interests: representations of p-adic groups, Deligne--Lusztig theory, local Langlands program, Fargues--Fontaine curve, v-stacks over schemes ...

News / Recent

• July 6-8, 2022: Maarten Solleveld will visit Bonn. He will give a talk on "The local Langlands program for non-supercuspidal representations" on July 7 at 14h (c.t.) in the Lipschitzsaal.
• May 3, 2022: Talk in the Arithmetic Seminar in Münster
• April 3-9, 2022: Oberwolfach Arbeitsgemeinschaft: Geometric Representation Theory, Link
• February 2022: My Heisenberg position (funded by DFG) will start at the University of Bonn.
• October 12, 2021: Talk at the Chinese University of Hong Kong. (Online)
• August 25, 2021: Talk at HCM Symposium in Bonn.
• April 16, 2021: Inaugural lecture at the University of Bonn. (In German, topic: "The sphere packing problem") The slides are available here.

Teaching

Im Sommersemester 2022 biete ich ein Seminar über lokale Körper an. Sollten Sie Interesse an einer Teilnahme haben, schreiben Sie mir eine E-Mail mit Ihrem Vortragswunsch.

Termin: Dienstags, 10(c.t.) - 12 (erster Vortrag am 12.4).
Ort: N 0.007 (Neubau).

Programm + Informationen finden sich hier.

Das Seminar wird Dienstags, 10(c.t.)-12 stattfinden. Der Ort wird noch bekanntgegeben.
Wenn Sie Fragen zum Seminar haben, schreiben Sie mir einfach eine E-Mail.

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.

Preprints

1. On a decomposition of $$p$$-adic Coxeter orbits
preprint 2021.
[ arXiv ]
Abstract We analyze the geometry of some $$p$$-adic Deligne--Lusztig spaces $$X_w(b)$$ introduced in this article attached to an unramified reductive group $${\bf G}$$ over a non-archimedean local field. We prove that when $${\bf G}$$ is classical, $$b$$ basic and $$w$$ Coxeter, $$X_w(b)$$ decomposes as a disjoint union of translates of a certain integral $$p$$-adic Deligne--Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms, and prove a loop version of Frobenius-twisted Steinberg's cross section.
2. 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.
3. Arc-descent for the perfect loop functor and $$p$$-adic Deligne--Lusztig spaces
preprint 2020, submitted. (old title: On ind-representability of $$p$$-adic Deligne-Lusztig spaces)
[ 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.
4. 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.
5. Reconstructing decomposition subgroups in arithmetic fundamental groups using regulators
preprint 2014, submitted.
[ arXiv ]

Published (or accepted) articles

1. The Drinfeld stratification for $${\rm GL}_n$$ (with Charlotte Chan)
Selecta Mathematica (New Ser.) 27, 50 (2021).
[ arXiv ] [ Journal ]
AbstractWe define a stratification of Deligne--Lusztig varieties and their parahoric analogues which we call the Drinfeld stratification. In the setting of inner forms of $$GL_n$$, we study the cohomology of these strata and give a complete description of the unique closed stratum. We state precise conjectures on the representation-theoretic behavior of the stratification. We expect this stratification to play a central role in the investigation of geometric constructions of representations of $$p$$-adic groups.
2. 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.
3. 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.
4. Affine Deligne-Lusztig varieties at infinite level (with Charlotte Chan)
Mathematische Annalen 380 (2021), 1801-1890
[ arXiv ] [ Journal ]

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

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

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

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

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

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

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

12. 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)

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.