Arithmetic Fundamental Group Research Articles

LOCAL SECTIONS OF ARITHMETIC FUNDAMENTAL GROUPS OFp-ADIC CURVES

AbstractWe investigatesectionsof the arithmetic fundamental group$\pi _1(X)$whereXis either asmooth affinoid p-adic curve, or aformal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, ifXadmits a compactificationY, and the exact sequence of$\pi _1(X)$ splits, then$\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of$\pi _1(X)$to arise from arational pointofY. One of the key ingredients in our investigation is the fact, we prove in this paper in caseXis affinoid, that the Picard group ofXisfinite.

Further Examples of Non-geometric Sections of Arithmetic Fundamental Groups

We show the existence of group-theoretic sections of certain geometrically pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves over p -adic local fields which are non-geometric , i.e., which do not arise from rational points. Among these quotients is the geometrically metabelian arithmetic fundamental group.

On the Semi-absoluteness of Isomorphisms between the Pro-$p$ Arithmetic Fundamental Groups of Smooth Varieties

Let p be a prime number. In the present paper, we consider a certain pro- p analogue of the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over p -adic local fields [i.e., finite extensions of the field of p -adic numbers \mathbb{Q}_p ] obtained by Mochizuki. This research was motivated by Higashiyama’s recent work on the pro- p analogue of the semi-absolute version of the Grothendieck conjecture for configuration spaces [of dimension \geq 2 ] associated to hyperbolic curves over generalized sub- p -adic fields [i.e., subfields of finitely generated extensions of the completion of the maximal unramified extension of \mathbb{Q}_p ].

The Semi-absolute Anabelian Geometry of Geometrically Pro-$p$ Arithmetic Fundamental Groups of Associated Low-Dimensional Configuration Spaces

Let p be a prime number. In the present paper, we study geometrically pro- p arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a p -adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro- p arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space.

Arithmetic of p-adic curves and sections of geometrically abelian fundamental groups

Let X be a proper, smooth, and geometrically connected curve of genus g(X)ge 1 over a p-adic local field. We prove that there exists an effectively computable open affine subscheme Usubset X with the property that {text {period}}(X)=1, and {text {index}}(X) equals 1 or 2 (resp. {text {period}}(X)={text {index}}(X)=1, assuming {text {period}}(X)={text {index}}(X)), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of Usplits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to X, and give a new characterisation of sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to {text {Pic}}^0 (resp. {text {Pic}}^{wedge }). As a consequence we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi [3] is orthogonal to {text {Pic}}^0.

The cuspidalisation of sections of arithmetic fundamental groups II

In this paper, which is a sequel to [7], we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i-th and 2/p-th step prosolvable arithmetic fundamental groups. As a consequence we exhibit two, necessary and sufficient, conditions for sections of arithmetic fundamental groups of hyperbolic curves over p-adic local fields to arise from rational points. We also exhibit a class of sections of arithmetic fundamental groups of p-adic curves which are orthogonal to Pic∧, and which satisfy (unconditionally) one of the above conditions.

Group-theoretic characterizations of almost open immersions of curves

Let p be a prime number and k either a finite field of characteristic p or a generalized sub-p-adic field. Let X1 and X2 be hyperbolic curves over k. In the present paper, we introduce a kind of morphism between X1 and X2 called an almost open immersion, and give some group-theoretic characterizations for the set of almost open immersions between X1 and X2 via their arithmetic fundamental groups. This result generalizes the Isom-version of Grothendieck's anabelian conjecture for curves over k which has been proven by S. Mochizuki and A. Tamagawa to the case of almost open immersions.

Notes on the local p-adic Simpson correspondence

We complete the theory of the local p-adic Simpson correspondence developed by Faltings for rational coefficients. It asserts that there is an equivalence between the category of small \(\mathbb {Q}_p\)-generalized representations of the geometric fundamental group and that of small \(\mathbb {Q}_p\)-Higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. The difficulty lies in the construction of the latter from the former, and we give it via a generalized Sen’s theory for the log smooth affine scheme, which depends on Faltings’ almost purity theorem. Inspired by a recent work by R. Liu and X. Zhu, we also give a formulation of local p-adic Simpson correspondence for \(\mathbb {Q}_p\)-generalized representations of the arithmetic fundamental group, and a characterization of Hodge-Tate generalized representations in terms of the correspondence.

A refined version of Grothendieck’s anabelian conjecture for hyperbolic curves over finite fields

In this paper we prove a refined version of a theorem by Tamagawa and Mochizuki on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one “ignores” the information provided by a “small” set of primes.

Godeaux–Serre varieties with prescribed arithmetic fundamental group

We show that for any given field k and natural number r≥2, every continuous extension of the absolute Galois group Galk by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective variety over k of dimension r.

On the Section Conjecture over Function Fields and Finitely Generated Fields

We investigate sections of the arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over \mathbb Q if it holds over all number fields, under the condition of finiteness (of the \mathcal l -primary parts) of certain Shafarevich–Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.

On the cuspidalization problem for hyperbolic curves over finite fields

In this article, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this article asserts that any Frobenius-preserving isomorphism between the geometrically pro-l fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-l fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.

Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki

These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on interuniversal Teichmuller theory [31], which might also be called arithmetic deformation theory, and its application to diophantine geometry. They provide an external perspective which complements the review texts [32] and [33]. Some important developments which preceded [31] are presented in the first section. Several important aspects of arithmetic deformation theory are discussed in the second section. Its main theorem gives an inequality–bound on the size of volume deformation associated to a certain log-theta-lattice. The application to several fundamental conjectures in number theory follows from a further direct computation of the right hand side of the inequality. The third section considers additional related topics, including practical hints on how to study the theory.

On the existence of non-geometric sections of arithmetic fundamental groups

We show the existence of group-theoretic sections of the “étale-by-geometrically abelian” quotient of the arithmetic fundamental group of hyperbolic curves over \(p\)-adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.

On the [formula omitted]-adic section conjecture

We prove that the analog of the Grothendieck anabelian section conjecture for curves holds true over a p-adic local field, if the following two assertions hold true. First, every section of the arithmetic fundamental group of a hyperbolic curve over a p-adic local field has an algebraic cycle class. Second, given a section of the cuspidally abelian absolute Galois group of the function field of such a curve, the corresponding section of the maximal Z/pZ-elementary abelian quotient, and a neighborhood of the latter, then the index of the corresponding curve is prime-to-p.

The cuspidalisation of sections of arithmetic fundamental groups

We prove that sections of arithmetic fundamental groups of hyperbolic curves with cycle classes uniformly orthogonal to Pic∧ can be lifted to sections of cuspidally abelian absolute Galois groups, under a certain condition on the base field (which is satisfied for instance by fields with finitely generated absolute Galois groups). We also observe the existence of sections of geometrically pro-Σ arithmetic fundamental groups of hyperbolic curves over p-adic local fields, p∉Σ, which do not arise from rational points.

Remarks on Cycle Classes of Sections of the Arithmetic Fundamental Group

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.

Virtual Betti numbers of compact locally symmetric spaces

We show that the virtual Betti number of a compact locally symmetric space with arithmetic fundamental group is either the Betti number of the compact dual or else is infinite.

The fundamental groupoid scheme and applications

We define a linear structure on Grothendieck's arithmetic fundamental group $\pi_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar k/k)$ to $\pi_1(X, x)$ with the existence of a neutral fiber functor on the category which linearizes it. When applied to Grothendieck section conjecture, it allows us to find a $k$-structure on the universal covering, and $k$-rational pro-points at finite and infinite distance which lift given $k$-rational points. Changes as compared to the first version: (some) typos corrected, english impoved (?), (2.6) better explained, Thm 3.2, 3) made more precise, correspondingly Cor. 4.6 made more precise as well. It equates conjugacy classes of sections with neutral fiber functors. Finally Thm 5.1 establishes a bijection between rational points and neutral fiber functors under the assumption that Grothendieck's conjecture is true.

A finiteness conjecture on abelian varieties with constrained prime power torsion

The pro-$\ell$ Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched `$\ell$-covers.' In many cases, the $\ell$-power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for $\ell$-covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.