24 research outputs found
A note on determinant functors and spectral sequences
The aim of this note is to prove certain compatibilities of determinant functors with spectral sequences and (co)homology thereby extending results of [3] and refining a description in [9]. It turns out that the determinant behaves as well as one would have expected in this regard, only that we were not able to find references for it in the literature. The results are crucial for descent calculations in the context of Iwasawa theory [12] or Equivariant Tamagawa Number Conjectures [4, 5, 6]
The GL_2 main conjecture for elliptic curves without complex multiplication
The main conjectures of Iwasawa theory provide the only general method known
at present for studying the mysterious relationship between purely arithmetic
problems and the special values of complex L-functions, typified by the
conjecture of Birch and Swinnerton-Dyer and its generalizations. Our goal in
the present paper is to develop algebraic techniques which enable us to
formulate a precise version of such a main conjecture for motives over a large
class of p-adic Lie extensions of number fields. The paper ends by formulating
and briefly discussing the main conjecture for an elliptic curve E over the
rationals Q over the field generated by the coordinates of its p-power division
points, where p is a prime greater than 3 of good ordinary reduction for E.Comment: 39 page
Estimating the growth in Mordell-Weil ranks and Shafarevich-Tate groups over Lie extensions
Let E/Q be an elliptic curve, p > 3 a good ordinary prime for E, and Kâ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the MordellâWeil rank and ShafarevichâTate group for E over a tower of extensions K â/â inside Kâ; we obtain lower bounds on the former, and upper bounds on the latterâs size
LOCALISATIONS AND COMPLETIONS OF SKEW POWER SERIES RINGS
Abstract. This paper is a natural continuation of the study of skew power series rings A = R[[t; Ï, ÎŽ]] initiated in [9]. We construct skew Laurent series rings B and show the existence of some canonical Ore sets S for the skew power series rings A such that a certain completion of the localisation AS is isomorphic to B. This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings
Wach modules, regulator maps and epsilon-isomorphisms in families
We prove the âlocal Δ-isomorphismâ conjecture of Fukaya and Kato [13] for certain crystalline families of GQp-representations. This conjecture can be regarded as a local analog of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [33]), of Benois and Berger for crystalline GQp-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline GQp-representations with respect to abelian p-adic Lie extensions of Qpâ . Nakamura [24, 25] has also formulated a version of Katoâs Δ-conjecture for affinoid families of (Ï,Î)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an Δ-isomorphism for families of trianguline (Ï,Î)-modules, depending on a fixed triangulation. Our results imply that this Δ-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [6] and following Kisinâs approach to the construction of potentially semi-stable deformation rings [18]
On the non-commutative Main Conjecture for elliptic curves with Complex Multiplication
In [7] a non-commutative Iwasawa Main Conjecture for elliptic curves over Q has been formulated. In this note we show that it holds for all CM-elliptic curves E defined over Q. This was claimed in (loc. cit.) without proof, which we want to provide now assuming that the torsion conjecture holds in this case. Based on this we show firstly the existence of the (non-commutative) p-adic L-function of E and secondly that the (non-commutative) Main Conjecture follows from the existence of the Katz-measure, the work of Yager and Rubinâs proof of the 2-variable main conjecture. The main issues are the comparison of the involved periods and to show that the (non-commutative) p-adic L-function is defined over the conjectured in (loc. cit.) coefficient ring. Moreover we generalize our considerations to the case of CMelliptic cusp forms