Related papers: On the birational section conjecture with local co…

We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the $p$-adic approximation…

For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to…

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we show that the surjectivity statement in…

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between…

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…

We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $\rho$ on…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…

Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up…

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic…

Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any \epsilon>0, we show that X contains O(B^{D+\epsilon}) rational points of…

We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant…

Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli space…

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the…

We consider smooth projective curves C/$\mathbb{F}$ over a finite field and their symmetric squares $C^{(2)}$. For a global function field $K/\mathbb{F}$, we study the $K$-rational points of $C^{(2)}$. We describe the adelic points of…

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an…

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this paper we provide a proof of this conjecture, generalizing previous work of…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…