Prof. Dr. Alexander Ivanov

Preprints

1. Meromorphic vector bundles on the Fargues--Fontaine curve (with Ian Gleason), preprint 2023>
[ arXiv ]

Abstract

We introduce and study the stack of meromorphic G-bundles on the Fargues--Fontaine curve. This object defines a correspondence between the Kottwitz stack \(\mathfrak{B}(G)\) and \(Bun_G\). We expect it to play a crucial role in comparing the schematic and analytic versions of the geometric local Langlands categories. Our first main result is the identification of the generic Newton strata of BunmerG with the Fargues--Scholze charts M. Our second main result is a generalization of Fargues' theorem in families. We call this the meromorphic comparison theorem. It plays a key role in proving that the analytification functor is fully faithful. Along the way, we give new proofs to what we call the topological and schematic comparison theorems. These say that the topologies of \(Bun_G\) and \(\mathfrak{B}(G)\) are reversed and that the two stacks take the same values when evaluated on schemes.

2. 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.

3. 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.

4. Reconstructing decomposition subgroups in arithmetic fundamental groups using regulators, preprint 2014, submitted
[ arXiv ]

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Published (or accepted) articles

5. Arc-descent for the perfect loop functor and \(p\)-adic Deligne--Lusztig spaces
   Journal für die reine und angewandte Mathematik (Crelle's Journal) Vol. 2023, no. 794, pp. 1--54 (2023)
   (old title: On ind-representability of \(p\)-adic Deligne-Lusztig spaces)
   [ arXiv ] [ Journal ]
   Abstract

   We 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.
6. On loop Deligne-Lusztig varieties of Coxeter type for inner forms of \({\rm GL}_n\) (with Charlotte Chan)
   accepted in Cambridge Journal of Mathematics.
   [ arXiv ]
   Abstract

   We 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.
7. The Drinfeld stratification for \({\rm GL}_n\) (with Charlotte Chan)
   Selecta Mathematica (New Ser.) 27, 50 (2021).
   [ arXiv ] [ Journal ]
   Abstract

   We 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.
8. Cohomological representations of parahoric subgroups (with Charlotte Chan)
   Representation Theory 25 (2021), 1-26.
   [ arXiv ] [ Journal ]
   Abstract

   Generalizing 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.
9. The smooth locus in infinite-level Rapoport-Zink spaces (with Jared Weinstein)
   Compositio Mathematica 156 (2020), No. 9, 1846-1872.
   [ arXiv ] [ Journal ]
   Abstract

   Rapoport-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.

10. Affine Deligne-Lusztig varieties at infinite level (with Charlotte Chan)
    Mathematische Annalen 380 (2021), 1801-1890
    [ arXiv ] [ Journal ]

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

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

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

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

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

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

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

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

Here are all my articles on arXiv.

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