K. Kramer, A family of semistable elliptic curves with large TateShafarevich groups, Proc. Amer. Math. Soc. 89 (1983), 379-386. 45 Home/Search Document Not in Database Summary Related Articles Check |
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The Cassels-Tate Pairing and the Platonic Solids - Fisher (1 citation) (Correct)
....= 1, form a single orbit under the action of PSL 2 (Z=mZ) In x1 we prove Theorem 1. Let K a number eld and let m = 3; 4 or 5. Then the Tate Shafarevich group of an elliptic curve over K may contain arbitrarily many elements of order m. Special cases of this result are proved in [CaVI] B o] [Kr], McG] L] and [F1] Our method is that used in [F1] Since X(m) P 1 there are in nitely many elliptic curves E=K with E[m] m Z=mZ. We write : E E 0 and b : E E 00 for the isogenies with kernel m and Z=mZ respectively. We may estimate the Mordell Weil rank using either the ....
K. Kramer, A family of semistable elliptic curves with large TateShafarevich groups, Proc. Amer. Math. Soc. 89 (1983), 379-386. 45
Invariants des courbes de Frey-Hellegouarch et grands groupes de.. - Nitaj (1998) (1 citation) (Correct)
....que X est fini pour toute courbe elliptique modulaire d efinie sur Q et de rang analytique 0 ou 1. D autre part, Cassels [4] et Bolling [3] ont montr e que X peut etre arbitrairement grand (voir aussi [11] pour 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 ....
K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevitch groups, Proc. Am. Math. Soc. 89 (1983), 379--386.
Quadratic Twists Of Modular Forms And Elliptic Curves - Ono, Papanikolas (2000) (Correct)
....with an even number of prime factors. 2) All of the prime factors of each m i are in SE . Theorem 2.1 may also be used to prove the existence of non trivial elements of TateShafarevich groups of elliptic curves. Regarding Tate Shafarevich groups, works by Bolling, Cassels, Kramer, and Rohrlich [Bo, Ca, Kr, R] yield a variety of results concerning the nontriviality of the 2 and 3 parts of Tate Shafarevich groups for families of elliptic curves. Less is known about the non triviality of p parts of X(E) for primes p # 5. In this direction, Wong [Wo] proved that infinitely many quadratic twists of X 0 ....
K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevich groups, Proc. Amer. Math. Soc. 89 (1983), 473-499.
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