Let C ⊆ P be an unramified nonspecial real space curve having many real branches and few ovals. We show that C is a rational normal curve if n is even, and that C is anM -curve having no ovals if n is odd.

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m , for some m ∈ Z +. As a consequence we show that a maximal curve of genus g = (q − ...

We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along curve. In particular, at arbitrary small neighborhood curve, meromorphic function is constant. This implies Picard group countably generated.

Let Γ be a congruence subgroup of SL(2,Z). The modular curve X(Γ) and its Jacobian variety J(Γ) are very important objects in number theory. For instance, the problem of determining all possible structures of (Q-)rational torsion subgroup of elliptic curves over Q is equivalent to that of determining whether the modular curves X1(N) have non-cuspidal rational points. Also, the celebrated theore...

— In the present paper, we prove the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field. This finitenessmay be regarded as an analogue for a once-punctured elliptic curve of the well-known finiteness of the set of torsion rational points of an abelian variety over a number field. In order to obtain the finiteness, we discuss the center of th...

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...

The Pythagorean hodograph (PH) curves are polynomial parametric curves whose hodograph (derivative) components satisfy the Pythagorean condition. Lots of research works had been done with PH curves relevant topics due that PH curves own many remarkable properties, e.g., its offset curves have exact rational representations. In this paper, ball curves with rational polynomial offset are studied....

Given number fields L ⊃ K, smooth projective curves C defined over L and B defined over K, and a non-constant L-morphism h : C → BL, we denote by Ch the curve defined over K whose K-rational points parametrize the L-rational points on C whose images under h are defined over K. We compute the geometric genus of the curve Ch and give a criterion for the applicability of the Chabauty method to fin...

Let Xns(11) denote the modular curve associated to the normalizer of a non-split Cartan subgroup of level 11; see [10, Appendix]. This curve has genus 1, is defined over Q and parametrizes elliptic curves with a certain level 11 structure. The interest of Theorem 1.1 lies in the fact that Xns(11) is isomorphic over Q to the curve E and that rational points (x, y) on E for which x/(xy − 11) is i...

This paper shows how to construct a rational Bezier model of a swept surface that interpolates N frames (i.e., N position/orientation pairs) of a fixed rational space curve c(s) and maintains the shape of the curve at all intermediate points of the sweep. Thus, the surface models an exact sweep of the curve, consistent with the given data. The primary novelty of the method is that this exact mo...