J. Gebel, H.G. Zimmer, Computing the Mordell-Weil Group of an Elliptic Curve over Q. CRM Proc. and Lect. Notes 4 ( Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
Invariants des courbes de Frey-Hellegouarch et grands groupes de.. - Nitaj (1998) (1 citation) (Correct)
....ann Gammax . Soit d efinie par : x) p N 2 x Gamma(x)L(x) Alors le signe w de l equation fonctionnelle (2 Gamma x) w(x) v erifie w = Gamma1) r , o u r est le rang de E 1 . Le calcul de w peut se faire a l aide des coefficients an de la s erie L, par la m ethode expos ee dans [12]. Le calcul de (x) a la pr ecision 10 Gammak , k 1, n ecesssite le calcul de m p N coefficients an , avec m max 4 5 3=2 N 1=4 ; 1 i k log 10 Gamma log(1 Gamma e Gamma= p N ) j : Si E 1 admet une r eduction lisse ou multiplicative en 2 (f 2 1 dans la table ....
....(voir [28] Pour E 1 , si pjB p (A p Gamma B p ) alors le type de Kodaira pour p est de la forme I n et q = 1. Dans le cas o u le signe de l equation fonctionnelle de E 1 (s; t) est 1, on peut calculer la valeur de L(1) a la pr ecision 10 Gammak a l aide de l expression (voir [5] ou [12]) L(1) 2 m X n=1 an n e Gamma2n= p N ; o u m v erifie : m p N 2 i 2 log 2 k log 10 Gamma log i 1 Gamma e Gamma2= p N jj : Le calcul de la valeur de L(1) devient donc assez long pour de grandes valeurs du conducteur. Ceci est la principale raison du choix de la limite ....
[Article contains additional citation context not shown here]
J. Gebel and H.G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over Q, CRM Proc. Lect. Notes 4 (1994), 61--83.
SIMATH - a computer algebra system for number theoretic.. - Zimmer Self-citation (Zimmer) (Correct)
....ways, e.g. via Manin s conditional algorithm, 2 descent via 2 isogeny or general 2 descent, 3 descent. The rst method, taking the conjectures of Birch and Swinnerton Dyer Shimura Taniyama Hasse Weil for granted, was worked out and implemented over K = Q by J. Gebel (see [12]) The algorithm works for curves E over Q of rank r 7, where the case of rank 7 already takes some extra e ort. We mention that in the cases of rank r 1 parts of the Birch and Swinnerton Dyer conjecture are a theorem by work of Coates Wiles, Rubin, Kolyvagin, Gross Zagier and Diamond, and ....
....on k. In [8] we developed and implemented in SIMATH a general procedure for computing all integral points on elliptic curves E over the rationals Q. The procedure relies on a method of Lang and Zagier and requires the knowledge of the rank and a basis of the Mordell Weil group E(Q) Since, by [12], the group E(Q) can be regarded as known, the method of Lang and Zagier is applicable. The crucial tools are the N eron Tate height h on E(Q) and elliptic logarithms. For computing all integral points in E(Q) the classical complex elliptic logarithms suce, but if one wants to compute also all ....
J. Gebel, H.G. Zimmer, Computing the Mordell-Weil Group of an Elliptic Curve over Q. CRM Proc. and Lect. Notes 4 (
Basic Algorithms for Elliptic Curves - Zimmer Self-citation (Zimmer) (Correct)
....curve de ned over an algebraic number eld K. Then, according to Theorem 2.1, the Mordell Weil group is nitely generated, and we wish to mention four algorithms for computing the rank r and a basis of the free part E fr (K) Z r of E over K: 4.2. 1) Manin s conditional algorithm (see [18], 34] 4.2.2) Special 2 descent via 2 isogeny (following Tate, see [57] 4.2.3) General 2 descent (following Birch and Swinnerton Dyer, 2] 5] 6] 54] 4.2.4) General 3 descent (following J. Quer, see [43] 4.2.1 Manin s algorithm. Let MK be the set of all places p of the number eld ....
....at the argument s = 1. Of course, it is a problem to numerically decide whether or not L ( E=K; s) is zero at s = 1. However, on assuming that L ( E=K; 1) 6= 0, one inserts the value r 0 = in the Manin bound, where L (r 0 ) E=K; 1) occurs in the expression for R 0 (cf. [18], 34] and tries to compute a basis of E(K) with height below the new Manin bound. If one does not succeed, one must have r and hence L ( E=K; 1) 0. Table 4 An elliptic curve E over Q of rank r = 7 E : Y 2 1641Y = X 3 168X 2 161X 8 i Generators P i Heights h(P i ) 1 ....
J. Gebel; H.G. Zimmer, Computing the Mordell-Weil Group of an Elliptic Curve over Q. In: \Elliptic Curves and Related Topics", ed. by H. Kisilevsky and M. Ram Murty. CRM Proc. and Lect. Notes, vol. 4, 61-83. Amer. Math. Soc., Providence, R.I. 1994.
Solving Elliptic Diophantine Equations: The General Cubic Case - Stroeker, de Weger (1999) (Correct)
No context found.
J. Gebel and H.G. Zimmer, "Computing the Mordell-Weil group of an elliptic curve over Q ", CRM Proc. & Lect. Notes Vol. 4 (1994), 61--83.
Online articles have much greater impact More about CiteSeer.IST Add search form to your site Submit documents Feedback
CiteSeer.IST - Copyright Penn State and NEC