On the Number of Points of Elliptic Curves over a Finite Field and a Problem of B. Segre

Several theorems are studied concerning the number of points of an elliptic curve with a Legendre form on a finite field, in order to analyse the distribution of regular and pseudoregular points in relation to a hyperbola in a finite affine plane.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

A formula for the grössencharacter of a parametrized elliptic curve

A formula for the grössencharacter of an elliptic curve with complex multiplication, in a family parametrized by modified Weierstrass functions or classical theta-functions, is given. The method is based on Shimura's Reciprocity Law for modular functions, and applies to Legendre, Jacobi, and Hesse curves. As an application, the conductors of the CM curves in these families are determined.     

Elliptic curve analogue of Legendre sequences

The Legendre symbol is applied to the rational points over an elliptic curve to output a family of binary sequences with strong pseudorandom properties. That is, both the well-distribution measure and the correlation measure of order k, which are evaluated by using estimation of certain character sums along elliptic curves, of the resulting binary sequences are “small”. A lower bound on the linear complexity profile of these sequences is also presented. Our results indicate that the behavior of such sequences is very similar to that of the Legendre sequences. Research partially supported by the Science and Technology Foundation of Putian City (No. 2005S04), the Open Funds of Key Lab of Fujian Province University Network Security and Cryptology (No. 07B005) and the Foundation of the Education Department of Fujian Province (No. JA07164). Author’s addresses: Department of Mathematics, Putian University, Putian, Fujian 351100, China; and Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, China     

Evans–Krylov Estimates for a nonconvex Monge–Ampère equation

Mathematische Annalen - We establish Evans–Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The...     

On Hypergeometric Functions and Generalizations of Legendre's Relation

We prove some convexity properties for a sum of hypergeometric functions and obtain a generalization of Legendre's relation for complete elliptic integrals. We apply these results to prove some inequalities for hypergeometric functions, incomplete beta-functions, and Legendre functions.     

Elliott's identity and hypergeometric functions

Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

On the isomorphism classes of Legendre elliptic curves over finite fields

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.     

Ultrasmooth Vectors of Singular Elliptic Differential Operators

Spaces of ultrasmooth vectors of some elliptic differential operators are described. In particular, elliptic operators with strong singular coefficients in the neighborhood of the boundary and generalized Legendre differential operators are considered.     

On Legendre curves in contact 3-manifolds

It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR ³ with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR ³ is more natural for the study of Legendre curves than the Euclidean metric.This work was done while the first author was a visiting scholar at Michigan State University.     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

Cyclotomic problem, Gauss sums and Legendre curve

In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.     

Frontal Partner Curves on Unit Sphere S2

In this study, by using moving frame along frontal of Legendre curve, we define frontal partner curves on unit sphere S². We give the relationships between curvatures of Legendre curves and frontal partner curves are strengthen by an example.     

On a Certain Sub-Riemannian Geodesic Flow on the Heisenberg Group

Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals.     

Galois theory, discriminants and torsion subgroup of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.     

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.     

A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.