Related papers: Counting Rational Points on Cubic Curves

Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell$, we derive a uniform upper bound $\exp(C \log B / \log \log B)$ for the number of…

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic…

We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$.…

Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…

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 give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.

We describe the Hilbert scheme components parametrizing lines and conics on the space of determinantal nets of conics, N. As an application, we use the quantum Lefschetz hyperplane principle to compute the instanton numbers of rational…

Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two…

We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of…

We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface $x_1^3 + p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1, p_2$ and…

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a…

Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we…

Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…

In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields…

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

The main result of this note is that there are at most seven rational points (including the one at infinity) on the curve C_A with the affine equation y^2 = x^5 + A (where A is a tenth power free integer) when the Mordell-Weil rank of the…

We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $\mathbb{Q}$-rational points on all modular curves in terms of the rational points on finitely many…