Related papers: A Lower Bound for the Canonical Height on Abelian …

Let $K$ and $k$ be $p$-adic fields. Let $L$ be the composite field of $K$ and a certain Lubin-Tate extension over $k$ (including the case where $L=K(\mu_{p^{\infty}})$). In this paper, we show that there exists an explicitly described…

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of $\mathbb{Q}$. In particular, we prove that every elliptic curve $E$ over $\mathbb{Q}$ has the weak Hilbert property of…

We give a function F(d,n,p) such that if K/Q_p is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the prime-to-p…

We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming…

Denis associated to each Drinfeld module M over a global function function field L a canonical height function, which plays a role analogous to that of the Neron-Tate height in the context of elliptic curves. We prove that there exist…

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…

We prove some general results on syzygies of smooth projective varieties with numerically trivial canonical line bundle. This allows to confirm several cases of Mukai's syzygies conjecture for finite quotients of abelian varieties in any…

Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in…

The canonical height associated to a polarized endomporhism of a projective variety, constructed by Call and Silverman and generalizing the N\'eron-Tate height on a polarized Abelian variety, plays an important role in the arithmetic theory…

We extend to the general codimension a lower bound for the essential minimum on abelian varieties found in a previous work, under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, ``up to an…

Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a…

We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…

The torsion anomalous conjecture states that for any variety V in an abelian variety there are only finitely many maximal V-torsion anomalous varieties. We prove this conjecture for V of codimension 2 in a product E^N of any elliptic curve…

For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, \phi)$ over it, we establish a metric extension result for sections of $L^{\otimes n}$ from a…

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle.…

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…

We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$. In particular, we provide explicit equations defining the Kummer variety $\mathcal…

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a…

Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are…