V.A. Kolyvagin, Finiteness of E( ) and sha(E= ) for a subclass of Weil curves, Izv. Akad. Nauk. SSSR Ser. Mat. 52 (3) (  Home/Search   Document Not in Database   Summary   Related Articles   Check

....s = 1 is outside the domain of absolute convergence of the in nite product used to de ne L(E= s) One consequence of this conjecture is that E( is nite if L(E= 1) is non zero. This was proved by Coates and Wiles in [CW] for elliptic curves with complex multiplication, and by Kolyvagin [Kol] for modular elliptic curves. Recently, a di erent proof for modular elliptic curves has been announced by K. Kato. Thanks to the breakthroughs of [W3] TW] and [Di2] the results of Kolyvagin and Kato are now unconditional for a very large class of elliptic curves. The Shimura Taniyama ....

V.A. Kolyvagin, Finiteness of E( ) and sha(E= ) for a subclass of Weil curves, Izv. Akad. Nauk. SSSR Ser. Mat. 52 (3) (

....development of better algorithms for computing the analytic and arithmetic invariants that are so intriguingly related by them. We now know that the rst and, up to a non zero rational factor, the second conjecture hold for modular elliptic curves over Q 1 in the analytic rank 0 and 1 cases (see [GZ, Ko, Wal1, Wal2]) Furthermore, a number of people have provided numerical evidence for the conjectures for a large number of elliptic curves; see for example [BGZ, BSD, Ca, Cr2] By now, our theoretical and algorithmic knowledge of curves of genus 2 and their Jacobians has reached a state that makes it possible ....

V.A. Kolyvagin, Finiteness of E(Q) and Sh(E;Q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 522-540. MR 89m:11056

....sa finitude pour toute courbe elliptique est encore conjecturale, mais, en 1987, Rubin [29] a donn e un exemple d une famille infinie de courbes elliptiques pour lesquelles X est fini. Ces courbes sont a multiplication complexe, et d efinies sur un corps quadratique imaginaire. En 1989, Kolyvagin [18] a montr e 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 ....

....(voir Remarque 2.1) et sont telles que leurs fonctions L v erifient L(1) 6= 0. Ces courbes sont donc de rang nul. Ainsi, d apr es les travaux de Kolyvagin, le rang de leurs groupes de Tate Shafarevich est fini et toutes ces courbes v erifient la conjecture de Birch et Swinnerton Dyer (voir [15] et [18]) Il en r esulte que tous les exemples list es dans ces tables ne d ependent d aucune conjecture. 2. Les courbes de Frey Hellegouarch Soient s et t deux entiers tels que (s; t) est sans facteurs carr es. Pour tout nombre premier pjst(s Gamma t) on attribue un couple (A p ; B p ) a l aide de ....

V.A. Kolyvagin, Finiteness of E(Q) and X(E=Q) for a subclass of Weil curves, Math. USSR, Izv. 32 (1989), 523--541.

....evidence can be given for conjecture 3.3. 12 EULER SYSTEMS AND REFINED CONJECTURES Let Z denote a ring in which the following are invertible: 1. The primes 2 and 3. 2. All primes p (r 1) 2. 3. All primes p such that Gal(Q(E p 1) Q) is not isomorphic to GL 2 (Z p ) The methods of Kolyvagin [Ko1,Ko2,Ko3] allow one to show: theorem 3.4. Suppose that S is a product of primes which are inert in K. Then parts 1 and 2 of conjecture 3.3 are true. A proof of this result is given in [D1] and [D2] In fact, more precise information can be derived about the order of vanishing of 0 S ; cf. D2] An ....

Kolyvagin, V.A., Finiteness of E(Q) and III(E=Q) for a subclass of Weil curves, Izv. Akad. Nauk. SSSR Ser Mat. 52 (1988), 522-540.

....the d quadratic twist of E E(d) dy 2 = x 3 ax b. Moreover, if E is an elliptic curve defined over a number field K, then let rk(E, K) denote the rank of the Mordell Weil group E(K) Similarly, let X(E,K) denote the Tate Shafarevich group of E K. By a celebrated theorem of Kolyvagin [Kol] and the modularity of E, 2) implies the widely held speculation that (5) # D : rk(E(D) Q) 0 #E X. Heath Brown confirmed (5) for the congruent number elliptic curves in [HB] and subsequent works by James, Vatsal and Wong [Ko, V, Wo] confirm this assertion for a variety of families of ....

V. Kolyvagin, Finiteness of E(Q) and X E/Q for a subclass of Weil curves (Russian), Izv. Akad. Nauk., USSR, ser. Matem. 52 (1988), 522-540.

....s) at s = 1. By the Birch Swinnerton Dyer conjecture (BSD) this should equal the rank of the Mordell Weil group E f (Q ) If the analytic rank is odd, then obviously L(f; 1) 0, and we expect E f (Q) to be infinite: indeed, this has been proved in the rank 1 case by Kolyvagin et al. see [11]) In the next subsection we will see how to determine whether or not L(f; 1) 0 in all cases. 5.3. The ratio L(f; 1) Omega Gamma f) The real periods of the rational newform f are the periods hfl; fi where fl 2 H 1 (X 0 (N) Z) these are all integer multiples of the least positive real ....

V. I. Kolyvagin, Finiteness of E(Q) and X E=Q for a subclass of Weil curves, Math. USSR Izvest. 32 (1989), 523--542.

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V.A. Kolyvagin, Finiteness of E(Q) and (E=Q) for a subclass of Weil curves, Izv. Akad. Nauk. SSSR Ser Mat. 52 (3) (

....NEW TYPE OF EULER SYSTEMS. I. D. Yu. Logachev 1 1. Introduction. 1.1. The purpose of the present paper is to prove some important steps of a natural program of generalization of Kolyvagin s proof of finiteness of Tate Shafarevich group of a Taniyama Weil elliptic curve E of analytic rank 0 or 1 ([K1], K2] referred below as K case ) to the case of other Shimura varieties. Some intermediate results of the present paper are of its own importance. For example, analogs of Eichler Shimura relations for Picard modular varieties X (4.4.1 and 4.4.3) apparently were never published. Euler systems ....

....(1.5) 4.1) for the definition) denoted by XP . Some of these steps are valid without any significant change for other types of Shimura varieties. Remaining steps of the proof are of purely technical nature and do not contain serious obstacles. We consider for simplicity only the case of rank 0 ([K1]) although a construction of Euler systems of order 2 is given in (5.5) Section 1.2 below contains the plan of the present paper together with the plan of realization of the first part of the program. For each step of this plan it is indicated is the obtained proof complete or partial, and can ....

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Kolyvagin V.A., Finiteness of E(Q) and Sh(E; Q) for a subclass of Weil curves. Math. USSR Izvestiya, 1989, v. 32, No. 3, p. 523 - 541

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V.A. Kolyvagin, Finiteness of E(Q) and III(E=Q) for a subclass of Weil curves, Izv. Akad. Nauk. SSSR Ser Mat. 52 (3) (

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