Examples of torsion points on genus two curves

We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples , , and .

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.     

On curves of constant torsion I

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.     

Sharp estimates of torsion of elliptic curves

The best possible estimates of torsion of the curvesy ²=x ³+s andy ²=x ³+rx over an arbitrary algebraic number field are given.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 503–508, April, 1998.     

Submersions and curves of constant geodesic curvature

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub‐Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. We describe a canonical extension of the sub‐Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.     

Equivalence of higher torsion invariants

We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer–Weiss–Williams torsion is proportional to the higher torsion of Igusa and Klein.     

Elliptic curves and their torsion subgroups over number fields of type (2, 2,

Suppose that E：y2=x(x+M)(x+N) is an elliptic curve, where M相似文献   

On the torsion of functionally graded anisotropic linearly elastic bars

The torsion of homogeneous, isotropic, linearly elastic cylindricalbars has been the subject of numerous investigations from theoretical,computational and applied viewpoints. It is well known thatin this case, the circular shaft is the only one whose cross-sectiondoes not involve a warping displacement in the axial direction.For an inhomogeneous, isotropic, circular bar, there is alsono warping of the cross-section provided that the shear modulusvaries only in the radial direction. For general anisotropicmaterials, torsion induces bending and vice versa. For thoseanisotropic materials with at least one plane of elastic symmetrynormal to the axial direction, pure torsion is possible. Forhomogeneous materials of this type and special inhomogeneousmaterials, it has been shown by various arguments that no warpingoccurs for bars of particular elliptical cross-section. Here,we provide a unified derivation of the foregoing results. Somediscussion of torsional rigidities is also given.     

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Suppose thatE: y ² =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) _tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) _tors themselves. The order of E( K)_tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)_tors = E( F) _tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.     

Compactified Jacobians of curves with spine decompositions

A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri’s moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Here we give a sufficient condition for when these two types of moduli spaces are equal. Eduardo Esteves is Supported by CNPq, Processos 301117/04-7 and 470761/06-7, by CNPq/FAPERJ, Processo E-26/171.174/2003, and by the Institut Mittag–Leffler (Djursholm, Sweden).     

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.     

Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space

We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give B?cklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.     

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.     

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pic^a(X),M ∃ Pic^b (X) with h¹(X, L)= h¹(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ _L,M :H ⁰(X,L)⊗H ⁰(X,M)→H ⁰(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.     

一类常宽"等腰梯形"

本文由对角线等于底边长的等腰梯形构造了一类新的常宽“等腰梯形”, 而著名的常宽凸集圆盘与Reuleaux 三角形为退化的特例. 我们还证明了关于这类常宽“等腰梯形” 面积的Blaschke-Lebesgue定理.