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

To be announced

2026-12-07T16:30:00+09:00

December 7, 2026	To be announced Yamaguchi Naganori, Tokyo Denki University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 16:30 · FR: 08:30 To be announced

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December 7, 2026

To be announced

Yamaguchi Naganori, Tokyo Denki University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 16:30 · FR: 08:30

To be announced

To the main page of the AHGT seminar.

To be announced

2026-10-05T15:30:00+09:00

October 5, 2026	To be announced Ishii shun, RIMS Kyoto University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 15:30 · FR: 08:30 To be announced

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October 5, 2026

To be announced

Ishii shun, RIMS Kyoto University, JP RIMS Kyoto (Building 15 Room 201) + Zoom · JP: 15:30 · FR: 08:30

To be announced

To the main page of the AHGT seminar.

The Mathieu group M23 is a Galois group over Q

2026-09-14T21:00:00+09:00

September 14, 2026	The Mathieu group M23 is a Galois group over Q Rachel Pries and Xiaoyu Huang, Colorado State University, US and Temple University, US RIMS Kyoto (Room 110) + Zoom · JP: 21:00 · FR: 14:00 Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over \(\mathbb{Q}\) during 1984--1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group \(M_{23}\), occurs as a Galois group over \(\mathbb{Q}\). In fact, we produce an explicit degree 23 polynomial with rational coefficients whose splitting field has Galois group \(M_{23}\) over \(\mathbb{Q}\). To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of \(M_{23}\). We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of \(\mathbb{Q}(t)\) with Galois group \(M_{23}\). Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an \(M_{22}\)-extension of \(\mathbb{Q}\). This is joint work with Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.
X. Huang, B. Jackson, K.-H. Lee, B. Poonen, R. Pries, S. Zhang. The Mathieu group \(M_{23}\) is a Galois group over \(\mathbb{Q}\). Preprint (2026) [ArXiV] M. Musty, S. Schiavone, J. Sijsling, and J. Voight, A database of Belyi maps, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Ser., vol. 2, Math. Sci. Publ., Berkeley, CA, 2019, pp. 375–392. G. Malle and B. H. Matzat, Inverse Galois theory, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2018.
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September 14, 2026

The Mathieu group M23 is a Galois group over Q

Rachel Pries and Xiaoyu Huang, Colorado State University, US and Temple University, US RIMS Kyoto (Room 110) + Zoom · JP: 21:00 · FR: 14:00

Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over \(\mathbb{Q}\) during 1984--1989.

We complete this program by proving that the last remaining sporadic group, the Mathieu group \(M_{23}\), occurs as a Galois group over \(\mathbb{Q}\). In fact, we produce an explicit degree 23 polynomial with rational coefficients whose splitting field has Galois group \(M_{23}\) over \(\mathbb{Q}\). To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of \(M_{23}\). We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of \(\mathbb{Q}(t)\) with Galois group \(M_{23}\). Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an \(M_{22}\)-extension of \(\mathbb{Q}\).

This is joint work with Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.

X. Huang, B. Jackson, K.-H. Lee, B. Poonen, R. Pries, S. Zhang. The Mathieu group \(M_{23}\) is a Galois group over \(\mathbb{Q}\). Preprint (2026) [ArXiV]
M. Musty, S. Schiavone, J. Sijsling, and J. Voight, A database of Belyi maps, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Ser., vol. 2, Math. Sci. Publ., Berkeley, CA, 2019, pp. 375–392.
G. Malle and B. H. Matzat, Inverse Galois theory, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2018.

To the main page of the AHGT seminar.

Anabelian Geometry in Yokohama 2026

2026-09-08T20:59:00+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

Journée arithmétique et géométrique du LAGA, Univ. Sorbonne PN, FR

2026-06-05T20:59:00+09:00