52 research outputs found
Selmer Groups over p-adic Lie Extensions I
Let be an elliptic curve defined over a number field . In this paper,
we study the structure of the -Selmer group of over -adic Lie
extensions of which are obtained by adjoining to the
-division points of an abelian variety defined over . The main focus
of the paper is the calculation of the \Gal(F_\infty/F)-Euler characteristic
of the -Selmer group of . 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
Let be an elliptic curve over with good supersingular
reduction at a prime and . We generalise the definition of
Kobayashi's plus/minus Selmer groups over to
-adic Lie extensions of containing
, using the theory of -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
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 -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
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
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