The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section the action semisimple Lie algebra on its dual using principal s-triple. Actually it is enough to have an “adapted pair” and indeed construction in Joseph Shafrir (2010) [25] works rather well co-adjoint algebraic, but not necessarily r...

Linear descriptor systems are governed by dynamical equations subject to algebraic constraints. In the one-dimensional case, where only depend on a single index, usually time, Weierstrass canonical form splits up state vector in two parts, causal part, running forward and non-causal backward. this paper linear time-invariant autonomous two-dimensions discussed condition existence of non-trivial...

The Weierstrass curve X is a smooth algebraic determined by the canonical form, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r positive integer, and each Aj polynomial in x with certain degree. It known that every compact Riemann surface has X, which birational to surface. form provides projection ϖr:X→P as covering space. Let RX:=H0(X,OX(∗∞)) RP:=H0(P,OP(∗∞)). Recently, we obtained explici...

Isogenies are the morphisms between elliptic curves, and are accordingly a topic of interest in the subject. As such, they have been wellstudied, and have been used in several cryptographic applications. Vélu’s formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, Vélu’s formulas only work for elliptic curves specified by a Weiers...

We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double coverings of curves of genus large enough. 1. Let X be a projective, irreducible, non-singular algebraic curve defined over an algebraically closed field k of characteristic p. Let n ≥ 1 be an integer and CX a canonical divisor of X. The pluricanonical linear system |nCX | defines a nondegenerate...

This paper proves that the Montgomery form elliptic curves that are cheaply transformable into short Weierstrass form by a simple change of variables (x, y) 7→ (x+α, y) (instead of a more general affine change of variables) are precisely the curves with B = 1. The points of order 2 and 4 on these curves are described, and it is observed that the x-coordinates of these points are consecutive fie...