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Let $K$ be a sub-$p$-adic field. We show that the functor sending a finite type $K$-scheme to its \'etale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved the analogous theorem for fields finitely generated over $\mathbb{Q}$. Our proof relies on Mochizuki's Hom-theorem in anabelian geometry, and a study of point-theoretic morphisms of fundamental groups of curves., Comment: 12 pages, comments welcome

We study effective global generation properties of projectivizations of curve semistable vector bundles over curves and abelian varieties., Comment: 15 pages, comments welcome!

We contemplate the range of convex Fujita numbers for minimal smooth projective surfaces according to their position in the Kodaira-Enriques classification., Comment: 27 pages, comments welcome!

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 \'etale fundamental groups. We extend Tamagawa's result by adding a purely topological criterion for maps to agree up to a power of Frobenius., Comment: 14 pages, comments welcome!

It was recently proven by Esnault, Shusterman and the second named author, that the \'etale 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 $k$., Comment: Final version, accepted for publication in Comptes Rendus Math\'ematique, https://comptes-rendus.academie-sciences.fr/mathematique. Besides various minor improvements, a new section "More general base fields" was added

We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension $d \geq 2$ and an arbitrary projective group as fundamental group., Comment: Title changed. Accepted version, to appear in Adv. Geom

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 for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations., Comment: 59 pages, comments welcome

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties., Comment: 41 pages, revised version following advice by the referee

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale fundamental groups., Comment: 18 pages

We study the problem $\#\mathrm{EdgeSub}(\Phi)$ of counting $k$-edge subgraphs satisfying a given graph property $\Phi$ in a large host graph $G$. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of $p$-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field $\mathbb{F}_p$ which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of $\#\mathrm{EdgeSub}(\Phi)$ for minor-closed properties $\Phi$, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we investigate the problems of modular counting of paths, cycles, forests and matroid bases. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime $p$., Comment: 49 pages, 4 figures

We construct vertex transitive lattices on products of trees of arbitrary dimension $d \geq 1$ based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher dimensional cubical version of Ramanujan graphs (optimal expanders)., Comment: 35 pages

We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^3 = Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder. The formula we get in degree $3$ is compatible with the Hardy-Littlewood heuristic on average, agrees with the prediction for $q \equiv 2 \pmod 3$ but shows anomalies for $q \equiv 1 \pmod 3$., Comment: minor changes

Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k which is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings., Comment: 33 pages, refereed version

We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties., Comment: Accepted for publication in Algebra & Number Theory

For every $p\geq 2$ we show that each finite $p$-group with an unmixed Beauville structure is part of a surjective infinite projective system of finite $p$-groups with compatible unmixed Beauville structures. This leads to the new notion of an unmixed topological Beauville structure on pro-finite groups. We further construct for $p \geq 5$ a new explicit infinite series of non-abelian $p$-groups that allow unmixed Beauville structures., Comment: 7 pages, to appear in: Monatshefte der Mathematik

Unlike in characteristic 0, there are no non-trivial smooth varieties over an algebraically closed field k of characteristic p>0 that are contractible in the sense of etale homotopy theory., Comment: 6 pages, accepted for publication in Algebra & Number Theory

For an abelian variety A over a number field k we discuss the divisibility in H^1(k,A) of elements of the subgroup Sha(A/k). The results are most complete for elliptic curves over Q., Comment: 25 pages, this is part II of the original manuscript arXiv:1106.4255v1 that was split in two parts. Part I is now arXiv:1106.4255v2. To appear in: Journal f\"ur die Reine und Angewandte Mathematik

We construct an infinite series of simply transitive irreducible lattices in PGL_2(F_q((t))) \times PGL_2(F_q((t))) by means of a quaternion algebra over F_q(t). The lattices depend on an odd prime power q = p^r and a parameter \tau\ in F_q^* different from 1, and are the fundamental group of a square complex with just one vertex and universal covering T_{q+1} \times T_{q+1}, a product of trees with constant valency q + 1. Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over F_q((t)) with ample canonical class, Chern ratio (c_1)^2/c_2 = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value \chi = 1: the surface is a non-classical fake quadric., Comment: 37 pages

The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve $X$ over a finitely generated field $k$ of characteristic $0$ and any non empty Zariski open $U \subset X$, every section of $\pi _1 (X, \bar x) \to \mathrm{Gal}_k$ lifts to a section of $\pi _1 (U,\bar x) \to \mathrm{Gal}_k$. We consider in this article the problem of lifting Galois sections to the intermediate quotient $ \pi_1^{cc}(U)$ introduced by Mochizuki. We show that when $k = \mathbb Q$ and $D=X\setminus U$ is an union of torsion sub-packets every Galois section actually lifts to $ \pi_1^{cc}(U)$. One of the main tools in the proof is the construction of torus torsors $F_D$ and $E_D$ over $X$ and the geometric interpretation $ \pi_1^{cc}(U) \simeq \pi _1 (F_D)$., Comment: v2 corresponds to the published, deeply revised and much shortened version. We think that v1, that contains the same results, could still be helpful to the reader preferring a self-contained text

A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is cuspidal. The proof relies on $GL_2(F_\ell)$-quotients of $\pi_1(U)$ for some open U of X. If k is totally real or imaginary quadratic, we prove that all birationally adelic, non-cuspidal Galois sections come from rational points as predicted by the section conjecture of anabelian geometry. As an aside we also obtain a strong approximation result for rational points on hyperbolic curves over Q or imaginary quadratic fields., Comment: Theorem C (and Section 7) of the original version have been deleted due to a gap in the proof. This is the journal version

This note explores the consequences of Koenigsmann's model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.

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 function field of the curve., Comment: 32 pages, 4 figures, with an appendix on the zoo of valuations for 2-dimensional semi local fields

For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q., Comment: 22 pages, changed title, the origional manuscript arXiv:1106.4255v1 was split in two parts according to Div-questions (part I) and div-questions (part II: Weil-Ch\^atelet divisible elements in Tate-Shafarevich groups II: On a question of Cassels)

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of families of curves over number fields, such that the birational section conjecture is true in a non-trivial way., Comment: 20 pages - in this second version the applications (section 3) are reorganised as "abelian" applications (section 3) and "non-abelian" applications (section 4)

We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension., Comment: 32 pages, with an appendix containing a letter from P. Deligne to D. Thakur

This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil restriction of scalars. In particular, the etale fundamental group of the Weil restriction is determined by means of a Shapiro Lemma for non-abelian group cohomology.

Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of the fundamental group exact sequence. We follow and refine Nakamura's approach towards these boundary sections. For example, we obtain a weak anabelian theorem for hyperbolic genus 0 curves over quite general fields including for example the maximal abelian extension of the rational numbers.

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 having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points., Comment: 17 pages

A family of proper smooth curves of genus $\geq 2$, parametrised by an open dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map $\pi_1(U) \to \pi_1(S)$ on fundamental groups is an isomorphism. The criterion of this note is actually more precise., Comment: 12 pages; revised version including a section on outer representations for families of curves

The geometric condition of T. Saito for trivial action of the wild monodromy of a smooth proper curve over the generic point of a trait is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi-projective smooth curves with etale boundary divisor., Comment: 14 pages

An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with \pi^{tame} of curves over a finitely generated field and its effect on the set of isomorphisms. Universal homeomorphisms are formally inverted., Comment: LaTeX, 10 pages, to appear in Compositio Mathematica