Related papers: Purely inseparable points on curves
We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.
We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.
An irreducible smooth projective curve over $\mathbb{F}\_q$ is called \textit{pointless} if it has no $\mathbb{F}\_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field…
In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This…
For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value…
In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.
For a plane curve, a point on the projective plane is said to be Galois if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. We present upper bounds for the number of Galois…
In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.
A point on a plane curve is said to be Galois (for the curve) if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. It is known that the number of Galois points is finite except…
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…
The number A(q) is the upper limit of the ratio of the maximum number of points of a curve defined over $\Fq$ to the genus. By constructing class field towers with good parameters we present improvements of lower bounds of A(q) for q an odd…
Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of…
We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by…
We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is…
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…
In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of…
We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.
The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5,…
We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field.…
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…