D. Goldfeld and L. Szpiro, Bounds for the order of the Tate-Shafarevich group, Compositio Math. 97 (1995), 71-87. Home/Search Document Not in Database Summary Related Articles Check |
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Elliptic curves, L-functions, and CM-points - Zhang (2002) (Correct)
....order of X(E) then every element in S(E) m] comes from a global point in E(F ) Interestingly, modulo the Birch and Swinnerton Dyer conjecture and Riemann Hypothesis, Goldefeld and Szpiro have proved that Szpiro s Conjecture 2.5 is equivalent to the following: Conjecture 2. 7 (Goldfeld Szpiro [31]) For any constant # 0, there is a constant C # (F ) such that #X(E) C # (E)N F Q (N) 13 3 L functions and modular forms In this section we want to study L functions and modular forms, and their relation to the arithmetic questions addressed in the last section. The basic references ....
D. Goldfeld and L. Szpiro, Bounds for the order of the Tate-Shafarevich group, Compositio Math. 97 (1995), 71-87.
The abc-CONJECTURE - Browkin (Correct)
....many K Gammarational points. N. Elkies [El] proved that Mordell s conjecture follows from the abc Gammaconjecture for algebraic number fields (see 3.1 below) 2.3.6. Class numbers of imaginary quadratic fields. Let Gammad be a negative fundamental discriminant. A. Granville and H. Stark [GS] proved that the uniform abc Gammaconjecture (see 3.2. below) implies the following estimation from below for the class number of the field Q ( p Gammad) h( Gammad) i 3 o(1) j p d log d X 1 ff ; where the sum runs over all reduced quadratic forms ffx 2 fixy fly 2 of ....
....infinitely many m 2 Z: 2.3.8. Elliptic curves. Let E be an elliptic curve defined over Q . Denote by N its conductor, and by Delta its minimal discriminant. Then the Szpiro conjecture (see [Oe] Sz] says that j Deltaj C( N 6 : Moreover there is the Goldfeld Szpiro conjecture (see [GS]) giving an estimate of the order of Tate Shafarevich group of the curve E # C( N 1=2 : It is known (see [GS] that both these conjectures are equivalent provided the Birch Swinnerton Dyer conjecture is true. Now let a; b 2 N be relatively prime integers, and let a b = c: We consider ....
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D. Goldfeld, L. Szpiro, Bounds for the order of the Tate--Shafarevich group, Compositio Math. 97 (1995), 71--87.
Invariants des courbes de Frey-Hellegouarch et grands groupes de.. - Nitaj (1998) (1 citation) (Correct)
.... 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 Mathematics Subject Classification. 11G05, 11G40, 11Y99. ....
.... peut s av erer tr es difficile si le conducteur est assez grand (voir partie 2) En utilisant la relation (2) le rapport (1) devient : 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 ....
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D. Goldfeld and L. Szpiro, Bounds for the order of the Tate-Shafarevich group, Compos. Math. 97 (1995), 71--87.
Visualizing elements in the Shafarevich-Tate group - Cremona, Mazur (1998) (2 citations) (Correct)
....real number) Thus, as Ram Murty has shown in [Mur] the ABC conjecture is equivalent to the statement that the upper conductor exponent of the modular degree (f(E) mE ) for semistable elliptic curves is 2. See also current publications of A. Granville in this regard. Also, Goldfeld and Szpiro [G S] have conjectured that the upper conductor exponent of the order of the Shafarevich Tate group (f(E) jX(E=Q)j) is 1=2. See also [de W] where it is shown (conditional on the Birch Swinnerton Dyer conjecture and the Riemann hypothesis for Rankin Selberg zeta functions associated to certain ....
Goldfeld, D., Szpiro, L.: Bounds for the order of the Tate-Shafarevich group, Comp. Math. 97 (1995) 71-87.
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