Related papers: Visible Points On Exponential Curves

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.

In this paper, we present a probabilistic algorithm to compute the number of $\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by…

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

We show that measures of irrationality on very general codimension two complete intersections and very general complete intersection surfaces are multiplicative in the degrees of the defining equations. This confirms some cases of a…

An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of…

We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.

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.

We consider the problem of finding curves of minimum pointwise-maximum curvature, i.e., curves of minimax curvature, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We reformulate the problem in…

In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of…

We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.

Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this setup, characterization and classification problems lead to the natural question: "Given two points on X, how…

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.

We extend the refined version of the Chabauty-Coleman bound on the number of rational points on a curve of genus g>1 to the case of bad reduction.

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fields

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, $\pi$) which is defined as the maximum number of rational points over F q…

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of…

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible.…