Integral Iwasawa theory of galois representations for non-ordinary primes

Mathematische Zeitschrift - In this paper, we study the Iwasawa theory of a motive whose Hodge–Tate weights are 0 or 1 (thence in practice, of a motive associated to an abelian variety) at a...     

Toward equivariant Iwasawa theory

 Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G _∞ , X _∞ the Galois groups of K _∞ /k and M _∞ /K _∞ , respectively, where K _∞ is the cyclotomic l-extension of K and M _∞ the maximal abelian S-ramified l-extension of K _∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G _∞ ]]-module X _∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when lł[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l. Received: 7 August 2001 / Revised version: 6 May 2002 We acknowledge financial support provided by NSERC. Mathematics Subject Classification (2000): 11R23, 11R27, 11R32, 11R33, 11R37, 11R42, 11S20, 11S23, 11S31, 11S40     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

Computations in non-commutative Iwasawa theory

We study special values of L-functions of elliptic curves over twisted by Artin representations that factor through a falseTate curve extension .In this setting, we explain how to compute L-functions and thecorresponding Iwasawa-theoretic invariants of non-abelian twistsof elliptic curves. Our results provide both theoretical andcomputational evidence for the main conjecture of non-commutativeIwasawa theory.     

Separable Banach space theory needs strong set existence axioms

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, comprehension, is needed to prove such basic facts as the existence of the weak- closure of any norm-closed subspace of . This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for sentences. En route to our main results, we prove the Krein-\v{S}mulian theorem in , and we give a new, elementary proof of a result of McGehee on weak- sequential closure ordinals.

     

Complex multiplication and parity in Iwasawa theory

We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa's λ-invariants for these representations over dihedral -extensions are even. Our approach is a specialization argument and does not make use of Neková?'s deformation-theoretic Cassels pairing, though Neková?'s theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur-Rubin and give several examples in the context of abelian varieties and modular forms.     

Toward equivariant Iwasawa theory, III

Let l be an odd prime number and K _∞/k a Galois extension of totally real number fields, with and K _∞/k _∞ finite, where k _∞ is the cyclotomic -extension of k. The ``main conjecture' of equivariant Iwasawa theory, as formulated in [RW2], is, up to its uniqueness statement, reduced to the existence of a nonabelian pseudomeasure whenever G _∞=G(K _∞/k) is an l-group and Iwasawa's μ-invariant vanishes. This follows from combining the validity of the conjecture in the maximal order case with special congruences. The main tool of proof is a generalization of the Taylor-Oliver integral group logarithm so that it applies to the setting of Iwasawa theory. We acknowledge financial support provided by NSERC and the University of Augsburg.     

Non-anticipative representations of Banach space valued gaussian processes with respect to brownian motion

Necessary and sufficient conditions are found for the equivalence of the measures associated with (i) a Banach space valued Gaussian process, with mean 0, and (ii) a Bach space valued Brownian motion. The notion of a non-anticipative representation of (i) with respect to (ii) is defined and in the case of equivalence of the measures it is shown that such a representation exists and has an explicit stochastic integral form which is invertible. Theorems of Ershov on absolute continuity of measures associated with diffusion processes are extended to Banach space. Applications to infinite-dimensional filtering are considered.     

Fredholm theory for pairs of closed subspaces of a Banach space

We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces.     

Iwasawa theory for elliptic curves at supersingular primes

We give a new formulation in Iwasawa theory for elliptic curves at good supersingular primes. This formulation is similar to Mazur’s at good ordinary primes. Namely, we define a new Selmer group, and show that it is of Λ-cotorsion. Then we formulate the Iwasawa main conjecture as that the characteristic ideal is generated by Pollack’s p-adic L-function. We show that this main conjecture is equivalent to Kato’s and Perrin-Riou’s main conjectures. We also prove an inequality in the main conjecture by using Kato’s Euler system. In terms of the λ- and the μ-invariants of our Selmer group, we specify the numbers λ and μ in the asymptotic formula for the order of the Tate-Shafarevich group by Kurihara and Perrin-Riou. Oblatum 17-VI-2002 & 2-IX-2002?Published online: 18 December 2002     

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Covering a Banach space

A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.