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 text presents an informal overview on how, in accordance with some deeply
rooted principles of the philosophy of Alexander Grothendieck concerning the practice
of mathematics, recent progress in anabelian arithmetic geometry has led to the interuniversal Teichmüller theory (IUT) of Mochizuki Shinichi. The new geometry of monoids
furnished by IUT may be understood as the result of a seminal encounter between
Grothendieck’s principle of resolving the tension between the discrete and continuous
realms, on the one hand, and 𝑝-adic Hodge theory and height theory, on the other. In
doing so, it opens a new research frontier that goes beyond the Grothendieck geometry
of rings-schemes by providing a unifying framework for Diophantine and anabelian
arithmetic geometry.
Lectures notes “Around the inverse Galois problem”
The inverse Galois problem asks whether any finite group can be realised as
the Galois group of a Galois extension of the rationals. This problem and its refinements
have stimulated a large amount of research in number theory and algebraic geometry
in the past century, ranging from Noether’s problem (letting X denote the quotient of
the affine space by a finite group acting linearly, when is X rational?) to the rigidity
method (if X is not rational, does it at least contain interesting rational curves?) and to
the arithmetic of unirational varieties (if all else fails, does X at least contain interesting
rational points?). The goal of the present notes, which formed the basis for three lectures
given at the Park City Mathematics Institute in August 2022, is to provide an introduction
to these topics.
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and theoretical aspects and the other applications to existing theories.
This is a short survey on the subject of my talk at the Hyper-JARCS Memorial
Conference for Professor ,Stefan Papadima held at the University of Tokyo in Dec 2019.
In this paper k always denotes a finite field of characteristic . A variety over k means a reduced scheme separated and of finite type over k. An étale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9]).
This document is a short user\rqs guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.
The descent method is one of the strategies allowing one to study the Brauer–Manin obstruction to the local–global principle and to weak approximation on varieties over number fields, by reducing the problem to “descent varieties”. Very recently, in his Park City lecture notes, Wittenberg formulated a “descent conjecture” for torsors under linear algebraic groups. The present article gives a proof of this conjecture in the case of connected groups, generalizing the toric case from the previous work of Harpaz–Wittenberg. As an application, we deduce directly from Sansuc’s work the theorem of Borovoi for homogeneous spaces of connected linear algebraic groups with connected stabilizers. We are also able to reduce the general case to the case of finite (étale) torsors. When the set of rational points is replaced by the Chow group of zero-cycles, an analogue of the above conjecture for arbitrary linear algebraic groups is proved.
For a fixed finite group G, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched G-covers of the projective line. The main focus is on determining whether components obtained by "gluing" two other components, both defined over a number field K, are also defined over K. The article presents a list of situations in which a positive answer is obtained. As an application, when G is a semi-direct product of symmetric groups or the Mathieu group M23, components defined over Q of small dimension (6 and 4, respectively) are shown to exist.
Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth
Dans la perspective de construire une théorie de Galois infinie pour les
extensions non extérieures, nous montrons dans ce texte que le monoïde des
endomorphismes d’une Z-extension intérieure filtrée s’identifie à la complétion
procentrale du groupe de ses automorphismes intérieurs. En particulier, ce
monoïde a une structure topologique naturelle qui fait de lui un espace complet
et totalement discontinu.
In order to build an infinite Galois theory for non-outer extensions, we show
that the monoïd of endomorphisms of a filtered inner Z-extension identifies with
the procentral completion of the group of its inner automorphisms. In particular,
this monoid has a natural topological structure for which it is a complete and
totally discontinuous space.
We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.
We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.
We develop a local model theory for moduli stacks of 2-dimensional non-scalar tame potentially Barsotti–Tate Galois representations of the Galois group of an unramified extension of Qp. We derive from this explicit presentations of potentially Barsotti–Tate deformation rings, allowing us to prove structural results about them, and prove various conjectures formulated by Caruso–David–Mézard.
We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.
A Nakano-type generic vanishing result is extended from compact Kähler manifolds to manifolds in Fujiki class C, so that smooth proper complex algebraic varieties are covered.
Making the motivic group structure on the endomorphisms of the projective line explicit
Let E be a complex elliptic curve and S a non-empty finite subset of E. We show the equality of two algebras of multivalued functions on E∖S: on the one hand, an algebra constructed using the functions Γ introduced in arXiv:1712.07089 out of string theory motivations; on the other hand, the minimal algebra of holomorphic multivalued functions on E∖S which is stable under integration, studied in arXiv:2212.03119; both algebras coincide with the algebra of multivalued holomorphic functions with unipotent monodromy on E∖S and moderate growth at the points of S.
Let X be a smooth projective curve of genus ≥2 over a number field. A natural variant of Grothendieck’s Section Conjecture postulates that every section of the fundamental exact sequence for X which everywhere locally comes from a point of X in fact globally comes from a point of X. We show that X/Q satisfies this version of the Section Conjecture if it satisfies Kim’s Conjecture for almost all choices of auxiliary prime p, and give the appropriate generalisation to S-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over Z[1/2].
In the first two parts of the series, we constructed stabilizer subtorsors of a ‘twisted Magnus’ torsor, studied their relations with the associator and double shuffle torsors, and explained their ‘de Rham’ nature. In this paper, we make the associated bitorsor structures explicit and explain the ‘Betti’ nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro-p versions of the ‘Betti’ group of the double shuffle bitorsor.
We give sufficient conditions for finiteness of linear and quadratic refined Chabauty–Kim loci of affine hyperbolic curves. We achieve this by constructing depth <=2 quotients of the fundamental group, following a construction of Balakrishnan–Dogra in the projective case. We also apply Betts’ machinery of weight filtrations to give unconditional explicit upper bounds on the number of S-integral points when our conditions are satisfied.
Let B3 be the Artin braid group on 3 strands and PB3 be the corresponding pure braid group. In this paper, we construct the groupoid GTSh of GT-shadows for a (possibly more tractable) version GT0 of the Grothendieck-Teichmueller group GT introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of GT and denote it by GTgen. The objects of GTSh are finite index normal subgroups N of B3 satisfying the condition N⊂PB3. Morphisms of GTSh are called GT-shadows and they may be thought of as approximations to elements of GTgen. We show how GT-shadows can be obtained from elements of GTgen and prove that GTgen is isomorphic to the limit of a certain functor defined in terms of the groupoid GTSh. Using this result, we get a criterion for identifying genuine GT-shadows.
The classifying element for quotients of Fermat curves
It was recently proven by Esnault, Shusterman and the second named author, that the étale fundamental group of a connected smooth projective variety over an algebraically closed field k is finitely presented. In this note, we extend this result to all connected proper schemes over k.
We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class, there is a first-order sentence that characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.
Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.
We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa’s result by adding a purely topological criterion for maps to agree up to a power of Frobenius.
Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus 2 case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.
In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.
In this paper, we prove that arbitrary hyperbolic curves over p-adic local fields admit resolution of nonsingularities [“RNS”]. This result may be regarded as a generalization of results concerning resolution of nonsingularities obtained by A. Tamagawa and E. Lepage. Moreover, by combining our RNS result with techniques from combinatorial anabelian geometry, we prove that an absolute version of the geometrically pro-Σ Grothendieck Conjecture for arbitrary hyperbolic curves over p-adic local fields, where Σ denotes a set of prime numbers of cardinality ≥ 2 that contains p, holds. This settles one of the major open questions in anabelian geometry. Furthermore, we prove –again by applying RNS and combinatorial anabelian geometry– that the various p-adic versions of the Grothendieck-Teichmüller group that appear in the literature in fact coincide. As a corollary, we conclude that the metrized Grothendieck-Teichmüller group is commensurably terminal in the Grothendieck-Teichmüller group. This settles a longstanding open question in combinatorial anabelian geometry.
The Grothendieck conjecture for hyperbolic curves over finite fields was solved affirmatively by Tamagawa and Mochizuki. On the other hand, (a “weak version” of) the Grothendieck conjecture for some hyperbolic curves over algebraic closures of finite fields is also known by Tamagawa and Sarashina. So, it is natural to consider anabelian geometry over (infinite) algebraic extensions of finite fields. In the present paper, we give certain generalizations of the above results of Tamagawa and Sarashina to hyperbolic curves over these fields. Moreover, we give a necessary and sufficient condition for algebraic extensions of finite fields to be (torally) Kummer-faithful in terms of their absolute Galois groups.
In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck conjecture(s) for affine hyperbolic curves over fields finitely generated over the prime field. First of all, we show the conjecture over finite fields. Next, we show the geometrically m-step solvable version of the Oda-Tamagawa good reduction criterion for hyperbolic curves. Finally, by using these two results, we show the conjecture over fields finitely generated over the prime field.
Anabelian geometry has been developed over a much wider class of fields than Grothendieck, who is the originator of anabelian geometry, conjectured. So, it is natural to ask the following question: What kinds of fields are suitable for the base fields of anabelian geometry?
In this paper, we consider this problem for higher local fields. First, to consider “anabelianness” of higher local fields themselves, we give mono-anabelian reconstruction algorithms of various invariants of higher local fields from their absolute Galois groups. As a result, the isomorphism classes of certain types of higher local fields are completely determined by their absolute Galois groups. Next, we prove that mixed-characteristic higher local fields are Kummer-faithful. This result affirms the above question for these higher local fields to a certain extent.
In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic p>0 to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic p>0 which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus 0.
We introduce a new obstruction to lifting smooth proper varieties from characteristic
p > 0 to characteristic 0. It is based on Grothendieck’s specialization homomorphism
and the resulting discrete finiteness properties of étale fundamental groups.
In the present paper, we give an explicit construction of differences of tame fundamental groups of certain non-isomorphic curves via finite quotients. In particular, our construction shows that the anabelian phenomena for curves over algebraically closed fields of positive characteristic can be understood by using not only entire tame fundamental groups but also certain finite quotients of them.
The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties).
B. Moonen proved that i) implies ii) in characteristic 0, using p-adic Hodge theory.
We show that an unconditional result lies behind this implication: the \it observability of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen’s result.
We also discuss another aspect of the Tate conjecture related to the transcendence of p-adic periods.
The AHGT program naturally interacts with the latest developments of various fiels of mathematics (e.g., low-dimensional and algebraic topology, algebraic geometry and number theory, quantum field theory,...).
A note on Real line bundles with connection and Real smooth Deligne cohomology
We define a Real version of smooth Deligne cohomology for manifolds with involution which interpolates between equivariant sheaf cohomology and smooth imaginary-valued forms. Our main result is a classification of Real line bundles with Real connection on manifolds with involution.
We show that every automorphism of the congruence completion of the extended mapping class group which preserves the set of conjugacy classes of procyclic groups generated by Dehn twists is inner and that its automorphism group is naturally isomorphic to the automorphism group of the procongruence pants complex. In the genus 0 case, we prove the stronger result that all automorphisms of the profinite completion of the extended mapping class group are inner.
We describe an algorithm for determining a minimal Weierstrass equation for hyperelliptic curves over principal ideal domains. When the curve has a rational Weierstrass point w0, we also give a similar algorithm for determining the minimal Weierstrass equation with respect to w0.
Twisted Heilbronn Virtual Characters
by
G. Yamashita.
In preprint,
Oct 2023
Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic
Putman and Wieland conjectured that if Σ →Σ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H1(Σ ;Q) under the action of lifts to Σ of mapping classes on Σ are infinite. We prove that this holds if H1(Σ ;Q) is generated by the homology classes of lifts of simple closed curves on Σ. We also prove that the subspace of H1(Σ ;Q) spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that H1(Σ ;Q) is generated by the homology classes of lifts of loops on Σ lying on subsurfaces homeomorphic to 3-holed spheres.
Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles
The category of topological spaces endowed with two marked points is equipped with two families Fn and Hn of functors to the category of abelian groups, indexed by a nonnegative integer n: namely, the functor Fn takes the object (X,x,y) to the quotient of Zπ1(X,x,y) by an abelian subgroup associated with the n+1-st power of the augmentation ideal of the group algebra Zπ1(X,x), and the functor Hn takes the same object to the n-th singular homology group of Xn relative to a subspace defined in terms of partial diagonals. We construct a family of natural transformations νn:Fn→Hn. We identify the natural transformation obtained by restricting νn to the subcategory of algebraic varieties with a natural equivalence due to Beilinson.