Jekyll2026-05-14T11:03:29+09:00https://ahgt.math.cnrs.fr/feed/seminar.xmlArithmetic & Homotopic Galois Theory IRN | SeminarThe LPP-RIMS Arithmetic & Homotopic Galois Theory IRN is a CNRS France-Japan Research Network on Galois covers & moduli spaces, Motivic & Geometric Galois representations, and Arithmetic anabelian geometry. On algebraic geometry over division rings2026-07-13T15:30:00+09:002026-07-13T15:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/07/13/Paran_AG_div
July 13, 2026

On algebraic geometry over division rings

Elad Paran, Open University of Israel, IL RIMS Kyoto (Room 110) + Zoom · JP: 15:30 · FR: 08:30

We shall survey recent developments concerned with foundational aspects of quaternionic algebraic geometry:

1. A quaternionic Nullstellensatz for the ring R of polynomials in n central variables over the quaternion algebra H, in both abstract form (due to the author and Alon) and explicit form (due to M. Aryapoor).
2. A theorem about the geometry of zero sets of polynomials in R: If a polynomial vanishes on all common zeros with commuting coordinates of a left ideal J in R , then it vanishes on all common zeros of J in H^n. This result confirmed a conjecture of Gori, Sarfatti and Vlacci.
3. Study of contraction properties of one-sided ideals in polynomial rings over division rings. In particular, we show that if M is a maximal left ideal in the polynomial ring D[x], where D is a division ring, then the contraction of M to D need not be maximal. This resolved a question of Amitsur and Small from 1978. We shall discuss connections between this question to recent works and the implications to algebraic geometric over division rings.
4. A Nullstellensatz for quaternionic polynomial functions, a generalization to arbitrary centrally finite division rings by Bao and Reichstein, and an extension of the Ax-Grothendieck theorem to polynomial functions over centrally finite division rings.

Joint works with Gil Alon, Adam Chapman.

  1. G. Alon, E. Paran. A central quaternionic Nullstellensatz, Journal of Algebra, Volume 574, 15 May 2021
  2. G. Alon, E. Paran. A quaternionic Nullstellensatz, Journal of Pure and Applied Algebra, Volume 225, Issue 4, April 2021
  3. A. Chapman, E. Paran. Amitsur-Small rings, Journal of Algebra, Volume 679, 1 October 2025

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]]>On the section conjecture for the toric fundamental group2026-06-01T15:30:00+09:002026-06-01T15:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/06/01/Bresciani_section
June 1, 2026

On the section conjecture for the toric fundamental group

Giulio Bresciani, Università di Pisa, IT RIMS Kyoto (Room 110) + Zoom · JP: 15:30 · FR: 08:30

The toric fundamental group is the Tannaka dual of a category of vector bundles which become direct sums of line bundles on a finite étale cover. It is an extension of the étale fundamental group scheme by a projective limit of tori.
Grothendieck's section conjecture for the étale fundamental group implies the analogous statement for the toric fundamental group. We call this the toric section conjecture. We prove that a resolution of the toric section conjecture would reduce the original one to particular cases about which more is known, mainly due to J. Stix.
We prove that abelian varieties over p-adic fields satisfy the toric section conjecture, and give strong evidence that it holds for hyperbolic curves over p-adic fields, too.

  1. G. Bresciani. On the section conjecture for the toric fundamental group, 2025 [ArXiv]

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]]>Perverse sheaves and the Shafarevich conjecture [rescheduled]2026-05-11T15:30:00+09:002026-05-11T15:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/05/11/Kraemer_perverse
May 11, 2026

Perverse sheaves and the Shafarevich conjecture [rescheduled]

Thomas Krämer, TU Chemnitz, DE RIMS Kyoto (Room 110) + Zoom · JP: 15:30 · FR: 08:30

The Shafarevich conjecture, a special case of the Lang-Vojta conjecture in Diophantine geometry, predicts that over any number field there only finitely many isomorphism classes of smooth projective canonically polarized varieties with given Hilbert polynomial and good reduction outside a given finite set of primes. For curves this was famously proven by Faltings on his way to the Mordell conjecture, but the higher-dimensional case remains wide open.

In the talk I will discuss joint work with Marco Maculan in which we prove the Shafarevich conjecture for a large class of varieties with globally generated cotangent bundle. We combine the Lawrence-Sawin-Venkatesh method with the big monodromy theorem from our work with Javanpeykar, Lehn and Maculan. The key input is the convolution of perverse sheaves on abelian varieties.

  1. T. Krämer and M. Maculan, The Shafarevich conjecture for varieties with globally generated cotangent. Preprint (2025) [ArXiV]
  2. T. Krämer and M. Maculan, Arithmetic finiteness of very irregular varieties. Duke Math. J. (to appear). Preprint (2025) [ArXiV]
  3. A. Javanpeykar, T. Krämer, C. Lehn and M. Maculan, The monodromy of families of subvarieties on abelian varieties. Duke Math. J. 174 (2025), 1045-1149 [ArXiV].

 To the main page of the AHGT seminar.

]]>Decidability problems for global and local fields2026-04-20T15:30:00+09:002026-04-20T15:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/04/20/Fehm-Decidability_pbs
April 20, 2026

Decidability problems for global and local fields

Arno Fehm, TU Dresden, Germany RIMS Kyoto Room 110 + Zoom · JP: 15:30 · FR: 08:30

Can one algorithmically decide whether a system of polynomial equations has a solution in the field of rational numbers? While this question is open, several closely related questions are by now answered. In particular, the answer is YES for local fields (with some caveats in positive characteristic), and NO for global fields if we ask for something only slightly stronger.

In this talk I will give an introduction to this area and will discuss several variations of the question, for different fields (global and local), different kinds of equations (polynomial, linear, ...), and with arithmetic extra conditions (absolute values, heights).

  1. S. Anscombe, P. Dittmann and A. Fehm. Axiomatizing the existential theory of Fq ((t)). Algebra & Number Theory 17(11):2013–2032, 2023
  2. J. Koenigsmann. Decidability in local and global fields. Proc. Int. Cong. of Math. – 2018, Rio de Janeiro, Vol. 1 (45–60)
  3. B. Poonen. Undecidability in number theory. Notices of the Amer. Math. Soc. 55(3), 2008.

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]]>Quasifibrations in étale homotopy theory2026-04-13T15:30:00+09:002026-04-13T15:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/04/13/Holzschuh_Etale_type
April 13, 2026

Quasifibrations in étale homotopy theory

Tim Holzschuh, IHES, FR RIMS Kyoto (Building 15 - Room 201) [map & Information] + Zoom · JP: 15:30 · FR: 08:30

Let S be a noetherian and normal scheme and \(f\colon X \to S\) a geometric fibration (e.g. a smooth and proper morphism) with geometrically connected fibres.

In [Fri73], Friedlander proved that, after completion away from char(S), the étale homotopy type of a geometric fibre of f coincides with the homotopy fibre of the induced map on étale homotopy types.

In this talk, I will explain a more conceptual proof (strategy) of Friedlanders result that works for arbitrary qcqs schemes S. This is joint work in progress with Alexander Schmidt and Jakob Stix.

  1. A. Dold, R. Thom. Quasifaserungen und unendliche symmetrische Produkte, Annals of Mathematics, 1958
  2. E. M. Friedlander. The etale homotopy theory of a geometric fibration, Manuscripta Mathematica, 1973
  3. P. J. Haine, T. Holzschuh, S. Wolf. Nonabelian basechange theorems and étale homotopy theory, Journal of Topology, 2024

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]]>On the relation between Grothendieck-Teichmüller, double shuffle, and Kashiwara-Vergne Lie algebras2026-03-09T16:30:00+09:002026-03-09T16:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/03/09/Alexeev_associators
March 9, 2026

On the relation between Grothendieck-Teichmüller, double shuffle, and Kashiwara-Vergne Lie algebras

Anton Alekseev, Genève University, CH RIMS Kyoto (Room 110) + Zoom · JP: 16:30 · FR: 08:30

The Grothendieck-Teichmüller (grt), double shuffle (dmr), and Kashiwara-Vergne (krv) Lie algebras encode universal infinitesimal symmetries of braided monoidal categories, of formal multiple zeta-values (MZVs), and of Lie algebras, respectively. They can all be realized as Lie subalgebras of derivations of the free Lie algebra with two generators, and conjecturally, they are all isomorphic to each other.

In the talk, we will recall definitions of these three Lie algebras, and we will review the status of isomorphism conjectures. In particular, we will state the fundamental result of Furusho on the injective Lie homomorphism from grt to dmr, on the injection of grt to krv, and a recent result by Enriquez-Furusho and Schneps on the injection of dmr to krv. We will also mention the results of Kuno and Ren on the emergent version of krv, and of Howarth-Ren, Enriquez-Furusho, and Markarian on characterization of dmr.

  1. D. Bar-Natan, On associators and the Grothendieck-Teichmuller group. I Selecta Math. (N.S.) 4 (1998), no. 2, 183–212.
  2. V. G. Drinfelʹd, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(\bar{Q}/Q). Leningrad Math. J. 2 (1991), no. 4, 829–860
  3. H. Furusho, Double shuffle relation for associators. Ann. of Math. (2) 174 (2011), no. 1, 341–360.
  4. A. Alekseev and C. Torossian. Kashiwara-Vergne conjecture and Drinfeld's associators. Ann. of Math. (2) 175 (2012), no. 2, 415–463.

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]]>Transfer principles for Galois cohomology and Serre's conjecture II2026-02-04T16:30:00+09:002026-02-04T16:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/02/04/Izquierdo_GaloisSerreII
February 4, 2026

Transfer principles for Galois cohomology and Serre's conjecture II

Diego Izquierdo, IMJ-PRG Université Paris Cité, France RIMS Kyoto (Room 110) + Zoom · JP: 16:30 · FR: 08:30

In this talk, I will present some transfer principles for the cohomological dimension of fields. I will then show some applications to Serre's Conjecture II, which predicts a characterization of cohomological dimension 2 fields in terms of rational points on torsors under algebraic groups. In particular, we will see that Serre's Conjecture II for characteristic zero fields implies the same conjecture for positive characteristic fields. This is joint work with Giancarlo Lucchini Arteche.

  1. P. Gille. Serre's conjecture II a survey. Quadratic forms, linear algebraic groups, and cohomology, Developments mathematics 18, p. 41-56, Springer, 2010 [preprint]
  2. D. Izquierdo, G. Lucchini Arteche. Homogeneous spaces, algebraic K-theory and cohomological dimension of fields, Journal of the European Mathematical Society, Vol. 24(6), p. 2169–2189, 2022.
  3. D. Izquierdo, G. Lucchini Arteche. Transfer principles for Galois cohomology and Serre's conjecture II. Advances in Mathematics, vol. 480, Part C, 110532, 2025 [ArXiV]

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]]>Perverse sheaves and the Shafarevich conjecture [Postponed]2026-01-19T16:30:00+09:002026-01-19T16:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2026/01/19/Kraemer_perverse
January 19, 2026

Perverse sheaves and the Shafarevich conjecture [Postponed]

Thomas Krämer, TU Chemnitz, DE RIMS Kyoto (Room 110) + Zoom · JP: 16:30 · FR: 08:30

The Shafarevich conjecture, a special case of the Lang-Vojta conjecture in Diophantine geometry, predicts that over any number field there only finitely many isomorphism classes of smooth projective canonically polarized varieties with given Hilbert polynomial and good reduction outside a given finite set of primes. For curves this was famously proven by Faltings on his way to the Mordell conjecture, but the higher-dimensional case remains wide open.

In the talk I will discuss joint work with Marco Maculan in which we prove the Shafarevich conjecture for a large class of varieties with globally generated cotangent bundle. We combine the Lawrence-Sawin-Venkatesh method with the big monodromy theorem from our work with Javanpeykar, Lehn and Maculan. The key input is the convolution of perverse sheaves on abelian varieties.

  1. T. Krämer and M. Maculan, The Shafarevich conjecture for varieties with globally generated cotangent. Preprint (2025) [ArXiV]
  2. T. Krämer and M. Maculan, Arithmetic finiteness of very irregular varieties. Duke Math. J. (to appear). Preprint (2025) [ArXiV]
  3. A. Javanpeykar, T. Krämer, C. Lehn and M. Maculan, The monodromy of families of subvarieties on abelian varieties. Duke Math. J. 174 (2025), 1045-1149 [ArXiV].

 To the main page of the AHGT seminar.

]]>Lifting of representations of Galois groups of local fields and of fundamental groups of surfaces2025-12-01T16:30:00+09:002025-12-01T16:30:00+09:00https://ahgt.math.cnrs.fr/seminar/2025/12/01/Demarche
December 1, 2025

Lifting of representations of Galois groups of local fields and of fundamental groups of surfaces

Cyril Demarche, IMJ-PRG France RIMS Kyoto (Room 111) + Zoom · JP: 16:30 · FR: 08:30

In a joint work with Andrea Conti and Mathieu Florence, we consider lifting questions for representations of profinite groups satisfying a two-dimensional Poincaré duality property: examples of such groups are absolute Galois groups of local fields, and (profinite completions of) topological fundamental groups of orientable compact surfaces. We prove in particular that any modulo \(p\) representation of such a group lifts to a representation modulo \(p^2\) (under an assumption about the roots of unity in the local field case), and that every unipotent representation modulo \(p\) lifts modulo \(p^r\), for all \(r\) (under a technical assumption about roots of unity in the arithmetic case). The strategy and the proof are as elementary as possible, and are based on Kummer theory and Poincaré duality.

  1. G. Böckle. Lifting mod p representations to characteristics \(p^2\), J. Number Theory 101 (2003), no. 2, 310–337.
  2. A. Conti, C. Demarche, M. Florence. Lifting Galois representations via Kummer flags, preprint 2025 [ArXiv],
  3. M. Emerton, T. Gee. Moduli stacks of étale (ϕ, Γ)-modules and the existence of crystalline lifts, Annals of Math. Studies, 2023.
  4. A. Merkurjev, F. Scavia. The lifting problem for Galois representations, preprint 2025 [Arxiv]

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]]>Malle's conjecture over function fields2025-11-06T21:00:00+09:002025-11-06T21:00:00+09:00https://ahgt.math.cnrs.fr/seminar/2025/11/06/Landesman_Malle
November 6, 2025

Malle's conjecture over function fields

Aaron Landesman, Harvard University, USA Zoom only (AiP25) · JP: 21:00 · FR: 13:00

The inverse Galois problem asks whether every finite group \(G\) can be realized as the Galois group of a field extension of the rational numbers. Malle's conjecture is a refined version of the inverse Galois problem which predicts the asymptotic number of such extensions. In joint work with Ishan Levy, we prove a version of Malle's conjecture, computing the asymptotic growth of the number of Galois \(G\) extensions of \(\mathbb F_q(t)\), for \(q\) sufficiently large and relatively prime to \(|G|\). We use tools from algebraic geometry to relate this conjecture to a question in topology about the cohomology of certain Hurwitz spaces. We then complete the proof by solving the topological question using techniques from homotopy theory.

  1. A. Landesman, I. Levy, Homological stability for Hurwitz spaces and applications, 2025 [ArXiv]
  2. A. Landesman, I. Levy, The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces, 2025 [ArXiv]
  3. A. Landesman, I. Levy, The stable homology of Hurwitz modules and applications, 2025 [ArXiv]

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