Related papers: On the slope of hyperelliptic fibrations with posi…

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform…

We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general we prove that…

We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to…

Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of…

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

We prove a sharp relative Clifford inequality for relatively special divisors on varieties fibered by curves. It generalizes the classical Clifford inequality about a single curve to a family of curves. It yields a geographical inequality…

We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of non-torsion rational points on elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that among the…

Let $f\colon X\to B$ be a semistable fibration where $X$ is a smooth variety of dimension $n\geq 2$ and $B$ is a smooth curve. We give the structure theorem for the local system of the relative $1$-forms and of the relative top forms. This…

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on…

For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds…

We show that Lang's hyperbolic and function version conjectures hold for surfaces $S$ of general type having a fibration of general type onto a curve $C$. The notion of multiplicity used is natural, but not classical, which leds to orbifold…

We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its…

We provide linear lower bounds for $f_\rho(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,\rho)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_\rho(L)$. This bound is…

We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results…

In this article, we extend the example constructed in the paper by Sormani-Tian-Wang to build new examples that satisfy the assumptions of the conjecture by Gromov. Each of these new examples of sequence converges to a limit space with…

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…

We prove, via an "arithmetic surjectivity" approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for…

In this paper, we investigate the general notion of the slope for families of curves $f: X \to Y$. The main result is an answer to the above question when $\dim Y = 2$, and we prove a lower bound for this new slope in this case over fields…

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the…

It is known that Szpiro's conjecture, or equivalently the ABC-conjecture, implies Lang's conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly…