Related papers: Lower bounds for heights on some algebraic dynamic…

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that…

A field F is said to have the Bogomolov Property related to a height function h, if h(a) is either zero or bounded from below by a positive constant for all a in F. In this paper we prove that the maximal algebraic extension of a number…

Let E --> C be an elliptic surface defined over a number field K. For each finite covering C' --> C defined over K, let E' --> C' be the pullback. We give a strong upper bound for the rank of E'(C'/K) in the case that C' --> C is an…

Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is…

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…

In this paper we improve the upper bound of the number $N_{K, n}(X)$ of degree $n$ extensions of a number field $K$ with absolute discriminant bounded by $X$. This is achieved by giving a short $\mathcal{O}_K$-basis of an order of an…

Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified…

Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such…

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\bar{K})$. We prove strong effective and uniform…

Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a positive lower bound C (depending on A, K, and L) for the…

Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$. More precisely,…

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that…

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$…

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than…

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex…

For points $(a,b)$ on an algebraic curve over a field $K$ with height $\mathfrak{h}$, the asymptotic relation between $\mathfrak{h}(a)$ and $\mathfrak{h}(b)$ has been extensively studied in diophantine geometry. When $K=\overline{k(t)}$ is…

Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of…

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. We formulate a question suggesting the optimal exponent…

A theorem of Tate asserts that, for an elliptic surface E/X defined over a number field k, and a section P of E, there exists a divisor D on X such that the canonical height of the specialization of P to the fibre above t differs from the…

The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an…