51 research outputs found

    Selmer Groups over p-adic Lie Extensions I

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    Let EE be an elliptic curve defined over a number field FF. In this paper, we study the structure of the pp^\infty-Selmer group of EE over pp-adic Lie extensions FF_\infty of FF which are obtained by adjoining to FF the pp-division points of an abelian variety AA defined over FF. The main focus of the paper is the calculation of the \Gal(F_\infty/F)-Euler characteristic of the pp^\infty-Selmer group of EE. The final section illustrates the main theory with the example of an elliptic curve of conductor 294.Comment: 23 page

    Signed Selmer Groups over p-adic Lie Extensions

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    Let EE be an elliptic curve over Q\mathbb{Q} with good supersingular reduction at a prime p3p\geq 3 and ap=0a_p=0. We generalise the definition of Kobayashi's plus/minus Selmer groups over Q(μp)\mathbb{Q}(\mu_{p^\infty}) to pp-adic Lie extensions KK_\infty of Q\mathbb{Q} containing Q(μp)\mathbb{Q}(\mu_{p^\infty}), using the theory of (ϕ,Γ)(\phi,\Gamma)-modules and Berger's comparison isomorphisms. We show that these Selmer groups can be equally described using the "jumping conditions" of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case.Comment: 21 page

    Rankin--Eisenstein classes in Coleman families

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    We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in pp-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.Comment: Updated version, to appear in "Research in the Mathematical Sciences" (Robert Coleman memorial volume

    Local epsilon isomorphisms

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    In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou regulator map, or its extension to the 2-variable case due to the first and third authors