Related papers: Infinitely many hyperelliptic curves with exactly …

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also…

We prove that a very general elliptic surface $\mathcal{E}\to\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\ge2$ contains no rational curves other than the section and components of singular fibers.…

In this paper, we construct an infinite family of elliptic curves whose rank is exactly two and the torsion subgroup is a cyclic group of order two or three, under the parity conjecture.

Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$ where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive cubes on $E$…

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any…

We use Iwasawa theory, at a prime $p$ inert in a quadratic imaginary field $K$, to study the arithmetic properties of mock plectic invariants for elliptic curves of rank two. More precisely, under some minor technical assumptions, we prove…

This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective)…

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Recently it was shown that the Diophantine equations describing such a cuboid…

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider…

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the…

We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the…

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational…

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate…

Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show…

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all…

We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is finite,…

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in…

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p =…

We generalize the group law of curves of degree three by chords and tangents to the Jacobi variety of a hyperelliptic curve. In the case of genus 2 we accomplish the construction by a cubic parabola. We derive explicit rational formulas for…