相关论文: Points of Low Height on Elliptic Curves and Surfac…

Let $E$ be an elliptic curve over $\F_p$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\F_p) |$. Let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm…

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results…

We study the minimum number of distinct distances between point sets on two curves in $R^3$. Assume that one curve contains $m$ points and the other $n$ points. Our main results: (a) When the curves are conic sections, we characterize all…

Gompf conjectured that the elliptic surface $E(n)_{p,q}$ has no handle decomposition without 1- and 3-handles. We prove that each of the elliptic surfaces $E(n)_{5,6}$, $E(n)_{6,7}$, $E(n)_{7,8}$ and $E(n)_{8,9}$ has a handle decomposition…

The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic…

Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results…

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show…

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which…

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and…

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4\chi(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch…

Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_E to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the natural…

A point $p$ on a smooth complex projective curve $C$ of genus $g>3$ is subcanonical if the divisor $(2g-2)p$ is canonical. In the moduli space of pointed curves, the subcanonical locus is described by pairs $(C,p)$ as above, and it consists…

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

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…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

Consider the elliptic curves given by $ E_{n,\theta}:\quad y^2=x^3+2s n x^2-(r^2-s^2) n^2 x $ where $0 < \theta< \pi$, $\cos(\theta)=s/r$ is rational with $0\leq |s| <r$ and $\gcd (r,s)=1$. These elliptic curves are related to the…