The Tate-Shafarevich Group for Elliptic Curves with Complex Multiplication II

Using Iwasawa theory, we establish some numerical results, and one weak theoretical result, about the enigmatic Tate-Shafarevich group of an elliptic curve defined over the rational field, with complex multiplication. These strengthen results of this kind proven in an earlier paper of ours.     

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

Let E/Q be an elliptic curve. For a prime p of good reduction,let E(F_p) be the set of rational points defined over the finitefield F_p. Denote by (#E(F_p)) the number of distinct prime divisorsof #E(F_p). For an elliptic curve with complex multiplication,the normal order of (#E(F_p)) is shown to be log log p. The normalorder of the number of distinct prime factors of the exponentof E(F_p) is also studied. 2000 Mathematics Subject Classification11N37, 11G20.     

The Iwasawa Main Conjectures for GL2

We prove the one-, two-, and three-variable Iwasawa-Greenberg Main Conjectures for a large class of modular forms that are ordinary with respect to an odd prime p. The method of proof involves an analysis of an Eisenstein ideal for ordinary Hida families for GU(2,2).     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

The ABC Conjecture Implies Vojta's Height Inequality for Curves

Following Elkies (Internat. Math. Res. Notices7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.     

平面曲线上几何MDS码的主猜想

本文用Ｌａｎｇ－Ｗｅｉｌ的一个经典结果证明了在一定维数限制下充分大域上平面代数曲线上ＭＤＳ码的主猜想成立。     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

The Selmer Groups of Elliptic Curves

Let E/K be an elliptic curve with K-rational p-torsion points.The p-Selmer group of E is described by the image of a map λk and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.     

Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

Vojta's Main Conjecture for blowup surfaces

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.

     

The Kato Conjecture for Elliptic Differential-Difference Operators with Degeneration in a Cylinder

Second-order elliptic differential-difference operators with degeneration in a cylinder associated with closed densely defined sectorial sesquilinear forms in L₂(Q) are considered. These operators are proved to satisfy the Kato conjecture on the square root of an operator.     

复环面情形的Suita猜想

对任意复环面的情形证明了推广的Suita猜想,即απK≥c~2(α∈R),其中c是修正后的对数容度,K是对角线上的Bergman核.还阐明了对任意亏格≥2的紧Riemann面情形的公开问题.文中结果的证明部分地依赖于椭圆函数理论.     

The subconvexity problem for GL2

Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL ₁ and GL ₂ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.     

Iwasawa Theory for Elliptic Curves at Unstable Primes

In this paper we examine the Iwasawa theory of modular elliptic curves E defined over Q without semi-stable reduction at p. By constructing p-adic L-functions at primes of additive reduction, we formulate a "Main Conjecture" linking this L-function with a certain Selmer group for E over the Zp-extension. Thus the leading term is expressible in terms of III_E, E(Q)_tors and a p-adic regulator term.     

Descent Calculations for the Elliptic Curves of Conductor 11

Let A be any one of the three elliptic curves over Q with conductor11. We show that A has Mordell–Weil rank zero over itsfield of 5-division points. In each case we also compute the5-primary part of the Tate–Shafarevich group. Our calculationsmake use of the Galois equivariance of the Cassels–Tatepairing. 2000 Mathematics Subject Classification 11G05, 11Y40,11R23.     

On the Dual Mean-Value Conjecture for Complex Polynomials

Mathematical Notes -