Arithmetic & Homotopic Galois Theory IRN

Special year ``Arithmetic Homotopy Geometry’’ at RIMS Kyoto, April 2027-March 2028. Three Seasons: with main conferences, introductory lectures, and workshops.

2027-04-01T20:59:00+09:00

Special year ``Arithmetic Homotopy Geometry’’ at RIMS Kyoto, April 2027-March 2028. Seasons A: Homotopy, rationality, and geometry

2027-04-01T20:58:00+09:00

Go Yamashita (RIMS Kyoto) visits IMJ-PRG Sorbonne

2026-08-24T20:59:00+09:00

On algebraic geometry over division rings

2026-07-13T15:30:00+09:00

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

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

To the main page of the AHGT seminar.

Niels Borne (IMJ-PRG) visits RIMS, Kyoto University

2026-06-26T20:59:00+09:00

On the section conjecture for the toric fundamental group

2026-06-01T15:30:00+09:00

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

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

To the main page of the AHGT seminar.

Perverse sheaves and the Shafarevich conjecture [rescheduled]

2026-05-11T15:30:00+09:00

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.
T. Krämer and M. Maculan, The Shafarevich conjecture for varieties with globally generated cotangent. Preprint (2025) [ArXiV] T. Krämer and M. Maculan, Arithmetic finiteness of very irregular varieties. Duke Math. J. (to appear). Preprint (2025) [ArXiV] 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].
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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.

T. Krämer and M. Maculan, The Shafarevich conjecture for varieties with globally generated cotangent. Preprint (2025) [ArXiV]
T. Krämer and M. Maculan, Arithmetic finiteness of very irregular varieties. Duke Math. J. (to appear). Preprint (2025) [ArXiV]
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.

Hommages à Hironaka Heisuke [Hommage de l’IHES]

2026-05-01T21:58:00+09:00

Anna Cadoret (IMJ-PRG) visits RIMS, Kyoto University (also Aug. 6 - Sept. 4, 2026)

2026-05-01T20:59:00+09:00

W. Porowski (RIMS Kyoto University) gives 3 lectures on ``the Grothendieck Section conjecture’’ [Prog]

2026-04-22T20:59:00+09:00