相关论文: Uniformity in Mordell-Lang for curves

In the previous paper, Hirakawa and the author determined the set of rational points of a certain infinite family of hyperelliptic curves $C^{(p;i,j)}$ parametrized by a prime number $p$ and integers $i$, $j$. In the proof, we used the…

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in…

In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from…

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary…

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is…

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 explain how one can efficiently determine the (finite) set of rational points on a curve of genus 2 over $\mathbb Q$ with Jacobian variety $J$, given a point $P \in J(\mathbb Q)$ generating a subgroup of finite index in $J(\mathbb Q)$.

Let $C$ be a smooth projective absolutely irreducible curve of genus $g \geq 2$ over a number field $K$ of degree $d$, and denote its Jacobian by $J$. Denote the Mordell--Weil rank of $J(K)$ by $r$. We give an explicit and practical…

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one.…

We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a…

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…

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide…

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as…

A smooth geometrically connected curve over the finite field $\mathbb{F}_q$ with gonality $\gamma$ has at most ${\gamma(q+1)}$ rational points. The first author and Grantham conjectured that there exist curves of every sufficiently large…