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On the Iwasawa theory of CM elliptic curves at super-singular primes
... Then In the situation that F ∞ contains the cyclotomic Z p -extension, the implication (A1) ⇒ (A2) has been observed in [5] and somewhat implicitly in [7,20,31]. When F ∞ is the anti-cyclotomic Z p of an imaginary field, such equivalence is also examined in [4,27]. ...
Let A be an abelian variety defined over a number field F with supersingular reduction at all primes of F above p. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of A over a p-adic Lie extension (not neccesasily containing the cyclotomic \Zp-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the p-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.
... The aim of this paper is to prove the conjecture 1.6, (equivalently the weak Jannsen conjecture) at ordinary and supersingular situation following the proofs given in the situation where w = 1, (Rubin [15] at ordinary primes, and Mc- Connel [11] at supersingular primes). We write in detail only the supersingular situation because the ordinary situation is known to the specialists. ...
We give a proof of the weak Leopoldt's conjecture à la Perrin-Riou, under some technical condition, for the p-adic realizations of the motive associated to Hecke characters over an imaginary quadratic field K of class number 1, where p is a prime >3 and where the CM elliptic curve associated to the Hecke character has good reduction at the primes above p in K. This proof makes use of the 2-variable Iwasawa main conjecture proved by Rubin. Thus we prove the Jannsen conjecture for the above p-adic realizations for almost all Tate twists.
In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the p-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of -torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soul\'e elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology. Comment: 60 pages, Latex2e