The two arithmetic universalities frameworks, (linear) motivic theory and (non-linear) anabelian homotopy theory, once pursued separately, now increasingly overlap: (a) nearly-abelian reconstructions (m-step reconstruction, abelian-by-central, section conjecture), (b) non-abelian Chabauty (pro-unipotent, m-step unipotent quotient, Selmer sections), and (c) monodromy method in local systems illustrate this new interface.
In these settings, derived perverse formalism, mixed Hodge structures, and p-adic Hodge techniques (esp. period maps) indicate special loci of interest for finer control of arithmetic properties; étale topological type provides an anabelian-motivic bridge for progress on both sides of the frontier.
Main Conference
Sept-Nov, 2027 RIMSThe homology-homotopy frontier in arithmetic geometry
Progress in arithmetic geometry can be viewed through the filter of two universalities in arithmetic: the motivic one, which serves as a backbone for the (co)homological approaches in terms of linear Galois representations, and the anabelian homotopic one, which exploits the arithmetic and the geometry of spaces, via the Grothendieck conjecture, in terms of the étale fundamental group. These two approaches are usually developed side-by-side for complementary perspectives, such as in Ihara's program. Recent progress has now sketched a newly permeable interface, as can be seen in:
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- Meta-abelian anabelian geometry, where anabelian reconstructions and invariants are reduced to their closest non-abelian form (e.g., m-step reconstructions for curves and fields, abelian-by-central approach, section conjecture).
- Non-abelian Chabauty theory, whose pro-unipotent component and its m-step unipotent quotients, stand at the motivic and nearly-abelian frontier (e.g., with Selmer sections, approximation of rational points, and mixed Tate motives via iterated integrals).
- The monodromy method in local systems and perverse sheaves (with applications to Regular Inverse Galois Theory and motivic theory), where variations in families are controlled by monodromy and the Galois homotopy symmetries are replaced by the Tannaka ones.
In many of these settings, p-adic Hodge theory, mixed Hodge structures, and period maps indicate special loci of interest inside moduli spaces (and some algebraic vs analytic frontiers). New formalisms shed further light on these loci: derived geometry brings finer control for singular locus and Galois representations properties; étale topological type provides an anabelian-motivic bridge to further progress on both sides.
The goal of this conference is to gather a panel of experts for further investigation and the development of an overarching perspective, where the explicit and gripping nature of motives on the one hand and the panoptic nature of anabelian-homotopic theory on the other will enrich each other.
Mini-courses
Non-abelian Chabauty method for motivic and rational approximation
Sept-Nov, 2027 RIMSLecturers: A. Betts (Cornell, US), D. Corwin (Ben Gurion, IL)
Inter-universal Teichmüller principles in arithmetic geometry [TBC]
Sept-Nov, 2027Lecturers: To be announced
Workshops
Arithmetic and motives
Sept-Nov, 2027 RIMSOrg.: B. Collas (RIMS Kyoto, JP), S. Kelly (Tokyo Univ., JP)
Arithmetic of local systems
Sept-Nov, 2027 RIMSOrg.: B. Collas (RIMS Kyoto, JP), Y. Yatagawa (IS Tokyo, JP)
Satellite Event
Arithmetic France-Japan 2027
Sept-Nov, 2027 TBAOrg.: B. Collas (RIMS Kyoto, JP), N. Imai (Tokyo, JP), T. Ochiai (IS Tokyo, JP)
※ Program details, exact event dates and registration information will be announced later. In the meantime, save the date and explore a selection of activities on related topics.