Lecture - Anabelian reconstructions principles

Arithmetic invariants from homotopy Galois theory

Introduction to the arithmetic of the étale fundamental group through its geometric Galois action. The lecture studies the example of the projective line minus three points and explains how inertia and decomposition groups can be reconstructed using Nakamura’s lemma.

Fields reconstruction à la Uchida

Presentation of Uchida-style field reconstruction from group-theoretic data. The lecture covers valuations, units, and tools from local and global class field theory (Brauer groups and Artin reciprocity) leading to multiplicative and then additive reconstruction.

Anabelian geometry for affine curves

Overview of Tamagawa’s approach to anabelian geometry of curves over number fields and finite fields. Key tools include semistable models, the Oda–Tamagawa good reduction criterion, Lefschetz-type counting of rational points, and multiplicative reconstruction of function fields.

Lean & Anabelian arithmetic

To be announced

Elements of absolute anabelian geometry

Introduction to techniques in absolute anabelian geometry inspired by Mochizuki. The lecture discusses Belyi cuspidalization, synchronization of cyclotomic data via Kummer theory, and applications of Hochschild–Serre spectral sequence together with Uchida’s lemma.

Mono-anabelian reconstruction over p-adic fields

Study of mono-anabelian reconstruction for p-adic fields using purely group-theoretic methods. The lecture presents principles, examples and counterexamples in mixed characteristic, and algorithms recovering numerical field-related invariants and cyclotomic data.

Anabelian properties of Hodge-Tate representations

Exploration of anabelian properties for p-adic Hodge theory. After introducing Hodge–Tate representations, the lecture explains absolute anabelian reconstruction for local p-adic fields and additive reconstruction via the ramification filtration.

References - Books

• L. Fu, Étale Cohomology Theory, Nankai Tracts in Mathematics, Vol. 14, 2015.
• A. Grothendieck; M. Raynaud, Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris), Vol. 3, Paris: SMF, 2003.
• J. Neukirch , A. Schmidt , K. Wingberg. Cohomology of Number Fields, Grundlehren der mathematischen Wissenschaften, Springer, 2008 -- esp. chapters (3,)6,7,8,9(,12).
• J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979.
• J. Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Springer, 1999.
• J.-P. Serre, Abelian ℓ-adic Representations and Elliptic Curves, W. A. Benjamin, Inc., 1968.

References - Research articles

• Y. Hoshi, Introduction to Mono-Anabelian Geometry, Publ. Math. Besançon, 2021.

• Y. Hoshi, A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields,Kodai Math. J. 36, no. 2, 2013.
• H. Nakamura, Rigidity of the arithmetic fundamental group of the punctured projective line,J. Reine Angew. Math. 405, 1990.
• S. Mochizuki, A Version of the Grothendieck Conjecture for p-adic Local Fields,Internat. J. Math. 8, 1997.
• S. Mochizuki, Topics in Absolute Anabelian Geometry I, II, & III,J. Math. Sci. Univ. Tokyo 19, 20, 22 (2012, 2013, 2015).
• K. Sawada, Algorithmic approach to Uchida’s theorem for one-dimensional function fields over finite fields, RIMS Kôkyûroku Bessatsu B84, 2021.
• A. Tamagawa, The Grothendieck Conjecture for affine curves, Compositio Math. 109, no. 2, 1997.
• K. Uchida, Isomorphisms of Galois Groups of Algebraic Function Fields,Annals of Mathematics, Vol. 106, no. 3, 1977.

Lectures - Variations around the Hilbert property

Lecturers: A. Behajaina (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (TU Dresden, DE), A. Javanpeykar (Radboud, NL), D. Neftin (Technion, IL)

Scientific committee: B. Collas (RIMS, JP), P. Dèbes (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (TU Dresden, DE) & S. Mochizuki (RIMS, JP).

Org.: S. Philip (Stockholm, SE), N. Yamaguchi (IS Tokyo, JP)