Elliptic curves x3 + y3 = D

Equality of distributions is shown of even and odd values of the order of the zero at the point s=1 of L-functions of elliptic curves x³+y³=D, where D is a positive integer not divisible by a cube.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 407–414, October, 1971.     

Construction of CM Picard curves

In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.

     

Plane valuations and their completions

We give explicit formulae to compute plane valuations. The main tool is a collection of expressions called Hamburger-Noether expan-sion. These expansions classify valuations and moreover give us some parametric equations for them.

We apply the above results to study the completion v of a plane valuation v. To do it we improve the above classification. This improvement affects the valuations associated to formal branches, since they are the only ones presenting differences among the usual invariants associated to v and to[vcirc]     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

某些椭圆曲线的L－级数的L（1）的非零性

本文考虑了椭圆曲线Γ＿Ｄ：Ｘ￣３＋Ｙ￣３＝ＤＺ￣３．以ＬＤ（ｓ）记Γ＿Ｄ的ＨｅｃｋｅＬ－级数．由Ｌ＿Ｄ（ｓ）的解析延拓我们将Ｌ＿Ｄ（１）展成有限项之和，然后通过建立ｙ￣２＝ｘ￣３－１６的一个处处有好的约化的模型，证明了当ｐ≡２或５（ｍｏｄ　９）时，Ｌ＿（ｐ￣２）（１）≠０．这些结果是对Ｂｉｒｃｈ和Ｓｗｉｎｎｅｒｆｏｎ－Ｄｙｅｒ猜想的支持。     

Elliptic functions and equations of modular curves

Let be a prime. We show that the space of weight one Eisenstein series defines an embedding into ${mathbb P}^{(p-3)/2}X_1(p)$ for the congruence group that is scheme-theoretically cut out by explicit quadratic equations. Received: 8 November 2000 / Published online: 17 August 2001     

Elliptic curves with a rational point of finite order

We try to find all the elliptic curves over Q, up to isomorphism, with non-degenerate reduction at all primes except 2 or 3 primes possessing a rational point of finite order by making use of some diophantine equations.     

Elliptic curves of odd modular degree

The modular degree m _E of an elliptic curve E/Q is the minimal degree of any surjective morphism X ₀(N) → E, where N is the conductor of E. We give a necessary set of criteria for m _E to be odd. In the case when N is prime our results imply a conjecture of Mark Watkins. As a technical tool, we prove a certain multiplicity one result at the prime p = 2, which may be of independent interest. Supported in part by the American Institute of Mathematics. Supported in part by NSF grant DMS-0401545.