MFO-RIMS tandem workshop: Arithmetic homotopy and Galois theory. Abstracts from the MFO-RIMS tandem workshop held September 24–29, 2023. (English) Zbl 1537.00019
Summary: This report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck’s original pillars, the present program has now evolved beyond the classical group-theoretic legacy to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry.
This panorama “closes the loop” by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara’s program and Nakamura-Tamagawa-Mochizuki’s anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations.
This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers.
This panorama “closes the loop” by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara’s program and Nakamura-Tamagawa-Mochizuki’s anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations.
This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers.
MSC:
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 11-06 | Proceedings, conferences, collections, etc. pertaining to number theory |
| 12-06 | Proceedings, conferences, collections, etc. pertaining to field theory |
| 14-06 | Proceedings, conferences, collections, etc. pertaining to algebraic geometry |
| 55-06 | Proceedings, conferences, collections, etc. pertaining to algebraic topology |
| 12F12 | Inverse Galois theory |
| 14G32 | Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) |
| 14H30 | Coverings of curves, fundamental group |
| 14H45 | Special algebraic curves and curves of low genus |
| 14F06 | Sheaves in algebraic geometry |
| 55Pxx | Homotopy theory |
| 14F22 | Brauer groups of schemes |
| 14G05 | Rational points |
| 14D15 | Formal methods and deformations in algebraic geometry |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
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| [219] | Quasi-supersingular finite flat commutative group schemes Motivated by Hoshi’s solution of the (generalized) Coleman conjecture in the case that J has superspecial (good) reduction (cf. [Hos22], Theorem E), we consider the following problem: Problem 1. Is the action of I K on X tors trivial if J has supersingular (good) reduction over K? We try to approach this problem in the same way as Hoshi’s strategy in the proof of [Hos22], Theorem E. We refer to this strategy as the [Hos22]-strategy, tentatively in this report. To explain the strategy, we prepare a few terminologies. Let n be a non-negative integer and G a p-torsion finite flat commutative group scheme over W of rank p 2n . Write G for the special fiber of G. Definition 1. |
| [220] | Let E be an elliptic curve over W . We shall say that E is supersingular if the special fiber of E is a supersingular elliptic curve over k. |
| [221] | We shall say that G (respectively, G) is superspecial if the following con-dition is satisfied: there exist supersingular elliptic curves E i /W (respec-tively, E i /k) (0 < i ≤ n) such that G ≃ ⊕ 0<i≤n E i [p] (respectively, G ≃ ⊕ 0<i≤n E i [p]). |
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