Cohomology and torsion cycles over the maximal cyclotomic extension

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with {\mathbf{Q}/\mathbf{Z}} -coefficients of a smooth proper {\overline{K}} -variety defined over K. We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

Wach Modules, Regulator Maps, and $\varepsilon$-Isomorphisms in Families

We prove the “local $\varepsilon$-isomorphism” conjecture of Fukaya and Kato [13] for certain crystalline families of $G_{{\mathbf{Q}_p}}$-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 $G_{{\mathbf{Q}_p}}$-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to abelian $p$-adic Lie extensions of ${\mathbf{Q}_p}$. Nakamura [24, 25] has also formulated a version of Kato’s $\varepsilon$-conjecture for affinoid families of $(\varphi,\Gamma)$-modules over the Robba ring, and proved his conjecture in the rank-$1$ case. He used this case to construct an $\varepsilon$-isomorphism for families of trianguline $(\varphi,\Gamma)$-modules, depending on a fixed triangulation. Our results imply that this $\varepsilon$-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].

Iwasawa Theory for K2n

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let $\mathematical script capital(O)_F_n\$ denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i ($\mathematical script capital(O)_F_n\$){p}.

NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES

Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

CONOYAUX DE CAPITULATION ET MODULES D'IWASAWA

Soit F un corps de nombres totalement réel et p un premier impair, on note K0 = F(ζp). Pour n ∈ ℕ, Kn désigne le n-ième étage de la ℤp-extension cyclotomique K∞/K0, An est la p-partie du groupe des classes de Kn, [Formula: see text] et N∞ est l'extension de K∞ obtenue en extrayant des racines p-primaires d'unités. Le but de cet article est de montrer que le dual de Pontryagin de la partie plus des conoyaux de capitulation [Formula: see text], sur laquelle l'action de Γ a été tordue une fois par le caractère cyclotomique et la partie moins de la ℤp-torsion du groupe de Galois Gal (N∞ ∩ L∞/K∞) sont isomorphes. Let F be a totally real number field and p an odd prime, we note K0 = F(ζp). For an integer n, Kn is the nth floor of the ℤp-cyclotomic extension K∞/K0, An is the p-part of the class group of Kn, [Formula: see text] and N∞ is the extension of K∞ generated by p-primary roots of units. In this article, we prove that the plus part of the capitulation's cokernel [Formula: see text], on which Γ-action was twisted on time by the cyclotomic character, and the minus part of the ℤp-torsion of the Galois group Gal (N∞ ∩ L∞/K∞) is isomorphic.

Characteristic elements, pairings and functional equations over the false Tate curve extension

We construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a primep≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of thep-adicL-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.