Related papers: On Grothendieck's section conjecture for curves of…

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve,…

Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for…

We investigate sections of 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 $\Bbb Q$ if it…

Every nontrivial abelian variety over a Hilbertian field in which the weak Mordell-Weil theorem holds admits infinitely many torsors with period any $n > 1$ which is not divisible by the characteristic. The corresponding statement with…

We study the structure of the \'etale fundamental groups of smooth curves over certain arithmetic schemes, and investigate the relative version of Grothendieck's anabelian conjecture in this setting. Consequently, every hyperbolic curve…

Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $\pi_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where…

The toric fundamental group is the Tannaka dual of a category of vector bundles which become direct sums of line bundles on a finite \'etale cover. It is an extension of the \'etale fundamental group scheme by a projective limit of tori.…

Let $X$ be a smooth projective curve of genus $\geq2$ 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…

In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension 2 over sub-p-adic fields. In 1-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-p-adic…

In the present paper, we show a new result on the geometrically $2$-step solvable Grothendieck conjecture for genus $0$ curves over finitely generated fields. More precisely, we show that two genus $0$ hyperbolic curves over a finitely…

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves, when $g\geq 3$ and the image of the $\ell$-adic cyclotomic character $G_k\to \Z^\times$ is infinite. Furthermore, we…

A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underline{\pi}_{1}(X)$ should somehow have essential dimension $1$. We…

We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the…

In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by…

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves…

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…

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite…

As already noted by Niels Borne and Michel Emsalem, there is a natural generalization of the section conjecture for proper orbicurves. Combined with the reformulation by Niels Borne and Angelo Vistoli of the conjecture in terms of the…

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic…

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a…