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 K∞ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions K ₙ/ₖ inside K∞; we obtain lower bounds on the former, and upper bounds on the latter’s 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; σ, δ]] 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 “local ε-isomorphism” 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 Qp⁠. Nakamura [24, 25] has also formulated a version of Kato’s ε-conjecture for affinoid families of (φ,Γ)-modules over the Robba ring, and proved his conjecture in the rank-1 case. He used this case to construct an ε-isomorphism for families of trianguline (φ,Γ)-modules, depending on a fixed triangulation. Our results imply that this ε-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 Kisin’s 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 Rubin’s 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