B. H. Gross, Kolyvagin's work on modular elliptic curves, in: L-function and Arithmetic (ed. J. Coates and M. J. Taylor) Cambridge University Press (1991).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Let = 1 denote the negative of the sign in the functional equation for L(E=Q; s) The following describes the action of on the Heegner points in E(H[n] p: Lemma 2.3 There exists 0 2 Gal(H[n] K) such that (n) 0 (n) in E(H[n] p. Hence, e (n) 0 )e (n) Proof: In [2], it is observed that (n) 0 (n) torsion; for some 0 2 Gal(H[n] K) By lemma 2.1, the group E(H[n] has no p torsion, and the rst statement follows. The second is a consequence of the identity: e = e ; which results from equation (2) Kolyvagin s idea is to construct ....

B.H. Gross, Kolyvagin's work on modular elliptic curves, Proc. Durham symposium on L-functions and arithmetic, 1989, to appear.

....2. The m Selmer group of J 0 is trivial, and hence J 0 (K) is nite. Sketch of Proof: To prove the rst part, observe that the hypothesis implies that the image of y K in E(K) pE(K) is non zero. The conclusion then follows from a more precise formulation of the theorem of Kolyvagin. cf. [Gr2], prop. 2.1) The second part follows from theorem 1.3 and the natural generalization (cf. BD2] of theorem 1.2 for eigenforms with non rational fourier coecients. In light of corollary 1.4, part 2 of proposition 1.5 is consistent with the Birch and Swinnerton Dyer conjecture. This proposition ....

B.H. Gross, Kolyvagin's work on modular elliptic curves. L- functions and arithmetic (Durham,

....Galois extension of Q which is unrami ed at p. We recall some standard facts on Heegner points over ring class elds of K. We do not strive for the greatest generality, but only state the results in the form which we shall need in the proofs. A more thorough discussion can be found in [Gr1] or [Gr4]. Let S be the set of square free integers prime to ND which are products of primes which are inert in K. For all T 2 S we are given the following data: 1. An abelian extension K T of K, the ring class eld of K associated to the order of conductor T . It is rami ed only at the places of K which ....

....These operators act on the eld K T and on the Mordell Weil group E(K T ) in the natural way. Given S 2 S and l a prime in S which is prime to S, let denote a prime of K above l and let ;S 2 S be the Frobenius automorphism associated to . Proposition 3. 10 N l ( Sl) a l (S) Proof: See [Gr4], p. 240, prop. 3.7. Proposition 3.11 (Sl) S (S) mod 0 ) where 0 is any prime of K Sl above . Proof: See [Gr4] p. 240, prop. 3.7. Propositions 3.10 and 3.11 make up the axioms of an Euler system for Heegner points in the sense of Kolyvagin [Ko3] The action of complex ....

[Article contains additional citation context not shown here]

B.H. Gross, Kolyvagin's work on modular elliptic curves, Proc. Durham symposium on L-functions and arithmetic, 1989, to appear.

....(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 ....

B.H. Gross, Kolyvagin's work on modular elliptic curves, in L-functions and arithmetic, Proc. Symp., Durham/UK

....in some detail. For the simplest case, where the Selmer group in question is the ideal class group of a real abelian number field and the (L) are constructed from cyclotomic units, see [29] For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19] 31] 30] [13]. 5.4. Wiles geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD (V n ) using Kolyvagin s method. This is the most technically difficult part of Wiles proof, and is the part of Wiles work he referred to as not yet complete, in ....

Gross, B., Kolyvagin's work on modular elliptic curves, in L-functions and arithmetic, London Math. Soc. Lecture Notes 153 (1991) 235--256.

....in [3] 40] the theorem in [3] is stated for Q, but its proof can be easily generalized to any number field) there is an E such that LE (f, s) has order equal to 1 at s = 1. It follows from Theorem C that z f has infinite order. Now the second part of Theorem A follows from Kolyvagin s method [17] [28] 29] 30] which applies directly to our case without any new di#culty. The only thing we need is to give a correct system of CM points which we will do at the end of this paper. Plan of proof. Now we sketch the proof of Theorem C. Let # and # be two cusp forms on GL 2 (A F ) of weight 2 ....

....B and C, it su#ces to prove the following generalization of a theorem of Kolyvagin: Proposition 7.2.1. Assume that the Heegner point y f in A is non torsion. Then . A(F ) has rank given by rankA(F ) O f : Z]ord s=1 L(s, f) X(A) is finite. In view of Kolyvagin s method for other cases [17] [28] 29] 30] we need only construct certain Euler system of CM points. We consider square free elements n # N F which are square free and prime to NDE and such that every prime factor # is inert in K. For every such n, we choose a CM point x n of the conductor n such that x n is included ....

B. H. Gross, Kolyvagin's work on modular elliptic curves, in: L-function and Arithmetic (ed. J. Coates and M. J. Taylor) Cambridge University Press (1991).

....which relates the central derivatives of certain Rankin L series and the heights of certain Heegner points on elliptic curves. Combined with Goldfeld s work on L series [14] this formula gives a solution to Gauss problem on class numbers; and combined with Kolyvagin s work on Euler system [16, 25], this formula gives the best evidence for the rank issue in the Birch and Swinnerton Dyer conjecture. In [17] Gross has proposed a program to generalize this formula to totally real elds with anticyclotomic characters. In our previous paper [31] we have worked out the program when the character ....

....= 1, and p j N = p is split in K. The case treated in our previous paper [31] is when F is totally real, is trivial, D; 2N) 1, and j N = is split in K. One immediate application of our Gross Zagier formula is to generalize the work of Kolyvagin Logachev and Bertolini Darmon [16, 25, 6] to obtain some evidence toward the Birch and Swinnerton Dyer conjecture in rank 1 case. The details will be given in later papers. Here we just notice y actually lives in some factor A whose L function is given by and its conjugates. Let Z[ be the subring of C generated by values ( ....

B. H. Gross, Kolyvagin's work on modular elliptic curves, in L-function and Arithmetic, J. Coates and M. J. Taylor, ed., Cambridge University Press, 1991, pp. 253-356.

..... Since q is odd, H i (T; Z=qZ) 2 ) 0 and the Hochschild45 Refined class number formulas for derivatives of L series 46 Serre spectral sequence H m (GL 2 (Z=qZ) T; H n (T; Z=qZ) 2 ) H m n (GL 2 (Z=qZ) Z=qZ) 2 ) 4:1) implies that H i (GL 2 (Z=qZ) Z=qZ) 2 ) 0. cf. [Gr4]) Taking the Galois cohomology of the short exact sequence 0 Gamma E q ( K) Gamma E( K) q Gamma E( K) Gamma 0 (4:2) gives the exact cohomology sequence, analogous to the Kummer sequence, which is the starting point for the descent formalism: 0 Gamma E(K) qE(K) Gamma H 1 (K; ....

....on the Mordell Weil group E(K T ) in the natural way. Given S 2 S and l a prime in S which is prime to S, let denote a prime of K above l and let oe ;S 2 Gamma S be the frobenius automorphism associated to . Proposition 4.2. 1 N l (y(Sl) a l y(S) The proof of this statement can be found in [Gr4]. Proposition 4.2.2 y(Sl) j oe ;S y(S) mod 0 ) where 0 is any prime of K Sl above . Proof: See [Gr4] Propositions 4.2.1 and 4.2.2 make up the axioms of an Euler system for Heegner points in the sense of Kolyvagin [Ko3] Complex conjugation acts on the Galois extension K T and on the ....

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B.H. Gross, Kolyvagin's work on modular elliptic curves, Proc. Durham symposium on L-functions and arithmetic, 1989, to appear.

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B. H. Gross, Kolyvagin's work on modular elliptic curves, in: L-function and Arithmetic (ed. J. Coates and M. J. Taylor) Cambridge University Press (1991).

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Gross, B., Kolyvagin's work on modular elliptic curves, in L-functions and arithmetic, London Math. Soc. Lecture Notes 153 (1991) 235-256. 60

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Gross, B.H., Kolyvagin's work on modular elliptic curves, Proc. Durham symposium on L-functions and arithmetic, Cambridge University Press, 1991, pp. 235-256.

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