Iwasawa theory for elliptic curves over imaginary quadratic fields

Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes

“…As already mentioned, the techniques employed in this paper are close to those of Bertolini [2]. Similar results could presumably be obtained via different approaches, for example by adapting the arguments of Ç iperiani in [7] (which rely on the techniques developed in [8]) or, following ), by using Λ-adic Kolyvagin systems as is done by Howard in [15].…”

“…As an application, when all the primes dividing the conductor of E split in K, we combine our main theorem with results of Ç iperiani and of Iovita-Pollack and obtain a "big O" formula for the Zp-corank of the p-primary Selmer groups of E over the finite layers of K∞/K that represents the supersingular counterpart of a well-known result for ordinary primes.2010 Mathematics Subject Classification. 11G05, 11R23.Heegner points that satisfy a kind of trace-compatibility relation of the sort needed to study restricted Selmer groups as in [2].As an application, combining our main theorem with results of Ç iperiani and of Iovita-Pollack, we obtain the following "big O" formula (Theorem 6.1) for the Z p -corank of the p-primary Selmer groups of E over the finite layers of K ∞ /K.This is the supersingular analogue of a well-known result for ordinary primes; in fact, Theorem 1.5 proves [3, Conjecture 2.1] when p is supersingular and K ∞ is the anticyclotomic Z p -extension of K. Here we would like to emphasize that, due to the failure of Mazur's control theorem in its "classical" formulation, knowledge of the Λ-corank of Sel p ∞ (E/K ∞ ) as provided by [7, Theorem 3.1] is not sufficient to yield the growth result described in Theorem 1.5 (see Remark 6.3 for more details). Moreover, assuming the finiteness of the p-primary Shafarevich-Tate group of E over K m for m ≫ 0, standard relations between Mordell-Weil, Selmer and Shafarevich-Tate groups of elliptic curves over number fields lead (at least when D = 1) to a formula (Corollary 6.5) for the growth of the rank of E(K m ).As already mentioned, the techniques employed in this paper are close to those of Bertolini [2].…”

“…11G05, 11R23.Heegner points that satisfy a kind of trace-compatibility relation of the sort needed to study restricted Selmer groups as in [2].As an application, combining our main theorem with results of Ç iperiani and of Iovita-Pollack, we obtain the following "big O" formula (Theorem 6.1) for the Z p -corank of the p-primary Selmer groups of E over the finite layers of K ∞ /K.This is the supersingular analogue of a well-known result for ordinary primes; in fact, Theorem 1.5 proves [3, Conjecture 2.1] when p is supersingular and K ∞ is the anticyclotomic Z p -extension of K. Here we would like to emphasize that, due to the failure of Mazur's control theorem in its "classical" formulation, knowledge of the Λ-corank of Sel p ∞ (E/K ∞ ) as provided by [7, Theorem 3.1] is not sufficient to yield the growth result described in Theorem 1.5 (see Remark 6.3 for more details). Moreover, assuming the finiteness of the p-primary Shafarevich-Tate group of E over K m for m ≫ 0, standard relations between Mordell-Weil, Selmer and Shafarevich-Tate groups of elliptic curves over number fields lead (at least when D = 1) to a formula (Corollary 6.5) for the growth of the rank of E(K m ).As already mentioned, the techniques employed in this paper are close to those of Bertolini [2]. Similar results could presumably be obtained via different approaches, for example by adapting the arguments of Ç iperiani in [7] (which rely on the techniques developed in [8]) or, following ), by using Λ-adic Kolyvagin systems as is done by Howard in [15].…”

“…Heegner points that satisfy a kind of trace-compatibility relation of the sort needed to study restricted Selmer groups as in [2].…”

Kolyvagin systems and Iwasawa theory of generalized Heegner cycles

“…First we observe that we only need to show thatΨ is injective, since it is surjective by Using the elements z f,n,α , one might be able to show that the Λ-module H ∞ is free of rank 1 by adapting the techniques developed by Bertolini in [1].…”