S. Lang, Conjectured diophantine estimates on elliptic curves, Progress in Math. 35, Birkhauser, (1983) Home/Search Document Not in Database Summary Related Articles Check |
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Elliptic curves, L-functions, and CM-points - Zhang (2002) (Correct)
....for step 2 is then . E#ectivity The theorem gives a beautiful description about the structure of E(F ) However, its proof does not provide an e#ective way to find all solutions, even though people believe there should be one as the following conjecture predicts: Conjecture 2. 2 (Lang [57]) Let E be an elliptic curve defined over a number field F of rank r. Then there exists a basis P 1 , P r for the free part of E(F ) satisfying C #,F N F Q (#E ) for all 1 r. Here #E is the minimal discriminant (explained later) C #,F is a constant depends on F and #. Of course, ....
S. Lang, Conjectured diophantine estimates on elliptic curves, Progress in Math. 35, Birkhauser, (1983)
Invariants des courbes de Frey-Hellegouarch et grands groupes de.. - Nitaj (1998) (1 citation) (Correct)
.... d autres r esultats) D autres m ethodes effectives ont et e d evelopp ees par Kramer [19] Mai et Murty [23] pour d eterminer des courbes elliptiques ayant un grand groupe de Tate Shafarevich, mais les premi eres estimations de l ordre de X ont et e conjectur ees par Manin et Lang (voir [24] et [21]) Dans cette direction, Goldfeld et Szpiro ( 14] ont propos e la conjecture suivante : Conjecture 1. Pour tout 0, il existe une constante C 1 ( 0 telle que si E=Q est une courbe elliptique de conducteur N et de groupe de Tate Shafarevich X, alors jXj C 1 ( N 1 2 : 1991 ....
.... : fl = 2 log i jT j 2 Omega C j log N 2 log i L (r) 1) Rr j log N : Si on admet l hypoth ese de Riemann pour la fonction L (voir par exemple [14] alors il existe une constante C 3 ( 0 telle que L (r) 1) C 3 ( N : D autre part, la conjecture de Lang (voir [21]) implique qu il existe une constante C 4 ( 0 telle que le r egulateur R v erifie R C 4 ( N Gamma ; si le rang r est born e. Ainsi, le rapport fl v erifie : fl 2 log i jT j 2 Omega C j log N 2 log i C3 ( C4 ( j log N 4 : Pour les courbes elliptiques ayant un ....
S. Lang, Conjectured diophantine estimates on elliptic curves, Prog. Math. 35 (1983), 155--172.
Universal Bound on the Performance of Lattice Codes - Tarokh, Vardy, Zeger (1999) (1 citation) (Correct)
....TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 2, MARCH 1999 for all nontrivial dimensional lattices, there exists a rate such that for all . To this end, we first recast the bound of (45) as (46) where is the coding gain defined in (2) Next, we make use of the following lemma, due to Hermite [21]. Lemma A.2: For any dimensional lattice , there exists a basis with (47) Dividing both sides of (47) by yields Since for all nonzero , we may further conclude that for (48) Now, consider the fundamental parallelotope (or the fundamental region) of the lattice spanned by the basis , that is, ....
S. Lang, "Conjectured Diophantine estimates on elliptic curves," in Arithmetic and Geometry, vol. I, M. Artin and J. Tate, Eds. Boston, MA: Birkh auser, 1983, pp. 155--171.
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