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Indice des unités elliptiques dans les ℤ p -extensions

Hassan Oukhaba (2007)

Bulletin de la Société Mathématique de France

Nous comparons le comportement dans les ℤ p -extensions du nombre de classes d’idéaux avec le comportement de l’indice du groupe des unités elliptiques de Rubin.

Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié (2012)

Journal de Théorie des Nombres de Bordeaux

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two primes 𝔭 and 𝔭 ¯ . Let k ∞ be the unique ℤ p -extension of k which is unramified outside of 𝔭 , and let K ∞ be a finite extension of k ∞ , abelian over k . Let 𝒰 ∞ / 𝒞 ∞ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of 𝒰 ∞ / 𝒞 ∞ are finite. Our approach uses distributions and the p -adic L -function, as defined in [5].

Iwasawa theory for elliptic curves over imaginary quadratic fields

Massimo Bertolini (2001)

Journal de théorie des nombres de Bordeaux

Let E be an elliptic curve over ℚ , let K be an imaginary quadratic field, and let K ∞ be a ℤ p -extension of K . Given a set Σ of primes of K , containing the primes above p , and the primes of bad reduction for E , write K Σ for the maximal algebraic extension of K which is unramified outside Σ . This paper is devoted to the study of the structure of the cohomology groups H i ( K Σ / K ∞ , E p ∞ ) for i = 1 , 2 , and of the p -primary Selmer group Sel p ∞ ( E / K ∞ ) , viewed as discrete modules over the Iwasawa algebra of K ∞ / K .

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Robert Harron, Antonio Lei (2014)

Journal de Théorie des Nombres de Bordeaux

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p -adic L -functions for the symmetric powers of f , thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus p -adic L -functions and prove an analogue of Pollack’s decomposition of the admissible p -adic L -functions....

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