Iwasawa Theory Research Articles

Non-abelian Pseudomeasures and Congruences between Abelian Iwasawa L-functions

The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.

Equivariant Iwasawa theory: an example

The equivariant main conjecture of Iwasawa theory is shown to hold for a Galois extension K/k of totally real number fields with Galois group an l -adic pro- l Lie group of dimension 1 containing an abelian subgroup of index l , provided that Iwasawa's \mu -invariant \mu(K/k) vanishes.

Congruences between abelian pseudomeasures

Following Deligne and Ribet (`Values of abelian $L$-functions at negative integers over totally real fields.' Invent. Math. 59 (1980), 227-286) we prove that the `torsion congruences' (as introduced in our paper `Non-abelian pseudomeasures and congruences between abelian Iwasawa $L$-functions.' To appear in Pure and Applied Mathematics Quarterly) hold and so reduce the `main conjecture' of equivariant Iwasawa theory to the integrality of the logarithmic pseudomeasure.

Plongements {\ell }-adiques et {\ell }-nombres de Weil

We define ℓ-adic analogs of classical Weil numbers in connexion both with complex or ℓ-adic imbeddings of number fields and real or ℓ-adic absolute values. As an application we give some consequences related to the Iwasawa theory of cyclotomic towers.

The algebraic [formula omitted]-adic [formula omitted]-function and isogeny between families of Galois representations

In this paper, we give a formula to compare the algebraic p -adic L -functions for two different lattices of a given family of Galois representations over a deformation ring R . This generalizes a classical comparison formula by Schneider [P. Schneider, The mu-invariant of isogenies, J. Indian Math. Soc. 52 (1987) 159–170] and Perrin-Riou [B. Perrin-Riou, Variation de la fonction L p -adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989) 347–358], for which R is the cyclotomic Iwasawa algebra. Recall that we studied the two-variable Iwasawa theory for residually irreducible nearly ordinary Hida deformations in [T. Ochiai, A generalization of the Coleman map for Hida deformations, The Amer. J. Math. 125 (2003) 849–892; T. Ochiai, Euler system for Galois deformation, Ann. Inst. Fourier 55 (2005) 113–146; T. Ochiai, On the two-variable Iwasawa Main Conjecture, Compos. Math. 142 (2006) 1157–1200] and we acquired sufficient understanding through these works. By applying our formula to Hida’s nearly ordinary deformations, we understand better the two-variable Iwasawa theory for residually reducible cases, where the choice of lattices is not unique anymore.

THE ANTICYCLOTOMIC MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES

The Main Conjecture of Iwasawa theory for an elliptic curve is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.

On Jannsen’s conjecture for Hecke characters of imaginary quadratic fields

We present a collection of results on a conjecture of Jannsen about the $p$-adic realizations associated to Hecke characters over an imaginary quadratic field $K$ of class number $1$. The conjecture is easy to check for Galois groups purely of local type (Section 1). In Section 2 we define the $p$-adic realizations associated to Hecke characters over $K$. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field $K$ at $p$, which is related to the property that a global Galois group is purely of local type. Without this regularity assumption at $p$, we present a review of the known situations in the critical case (Section 3) and in the non-critical case (Section 4) for these realizations. We relate the conjecture to the non-vanishing of some concrete non-critical values of the associated $p$-adic $L$-function of the Hecke character. Finally, in Section 5 we prove that the conjecture follows from a general conjecture on Iwasawa theory for almost all Tate twists.

Pro-$p$ Extensions of Global Fields and pro-$p$ Groups

The meeting Pro- p Extensions of Global Fields and pro- p Groups was organised by Nigel Boston (Madison), John Coates (Cambridge) and Fritz Grunewald (Düsseldorf). As the name of the meeting conveys, a primary aim was to bring together group theorists working in the field of pro- p groups and number theorists interested in pro- p extensions of global fields. The workshop consisted of over 25 talks, supplemented by informal discussions. Topics included: Galois groups of extensions with restricted ramification, self-similar and automata groups, non-commutative Iwasawa theory, groups acting on rooted trees. This meeting was well attended with over 50 participants; more than 30 of these came from countries other than Germany. The range of topics and the diverse backgrounds of the participants led to a stimulating exchange of recent results, challenging problems and general ideas. The organisers and participants thank the Mathematisches Forschungsinstitut Oberwolfach for providing the setting for this successful workshop. The following extended abstracts appear in the chronological order of the talks; they were collected and edited by Benjamin Klopsch (Düsseldorf).

Iwasawa theory and the Eisenstein ideal

We verify, for each odd prime p<1000, a conjecture of W. G. McCallum and R. T. Sharifi on the surjectivity of pairings on p-units constructed out of the cup product on the first Galois cohomology group of the maximal unramified outside p extension of Q(μp) with μp-coefficients. In the course of the proof, we relate several Iwasawa-theoretic and Hida-theoretic objects. In particular, we construct a canonical isomorphism between an Eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical Iwasawa module over an abelian pro-p Kummer extension of the cyclotomic Zp-extension of an abelian field. This Kummer extension arises from the Galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves which was considered by M. Ohta in order to give another proof of the Iwasawa main conjecture in the spirit of that of B. Mazur and A. Wiles. In turn, we relate the Iwasawa module over the Kummer extension to the quotient of the tensor product of the classical cyclotomic Iwasawa module and the Galois group of the Kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower. We give an application to the structure of the Selmer groups of Ohta's modular representation taken modulo the Eisenstein ideal

Non-abelian Iwasawa theory of ℤp-extensions

Non-abelian Iwasawa theory of ℤp-extensions

Anticyclotomic main conjectures for CM modular forms

Generalizing work of Agboola-Howard on CM elliptic curves, we prove a version of the main conjecture of Iwasawa theory over the anticyclotomic ℤp-extension of a quadratic imaginary field K for Selmer groups attached to modular forms with complex multiplication by K. We deduce a statement to the effect that if the L-function of a CM modular form f of even weight has odd order of vanishing at its central point, then a Selmer group attached to f over ℚ is infinite.

Asymptotics of coinvariants of Iwasawa modules under non-normal subgroups

Let G be a pro-p p-adic analytic group, thought of as a closed subgroup of GLN (Zp), and let Σ be a closed subgroup of G. Write Λ for the completed group algebra Zp[[G]] and let M be a finitely generated Λ-module. Let G = G ⊃ G ⊃ G ⊃ . . . be the descending sequence of principal congruence subgroups of G; write Gn for the quotient G/G and Σn for the image of Σ in Gn. Write Mn for the coinvariant quotient of M under G. Then Mn is a module for the group algebra Zp[Gn]. In Iwasawa theory, one often finds that the growth of arithmetic invariants of interest (e.g. class numbers, Mordell-Weil ranks) is controlled by a Λ-module M . In particular, the growth can be related to the Zp-ranks of the coinvariant quotients ofM by various subgroups of G. Understanding these quotients is a purely algebraic problem. For instance, Harris [3, Theorem 1.10] shows that, if M is a Λ-torsion module, rankZp Mn = O(p n(dimG−1)). (1)

Ranks of elliptic curves in cubic extensions

For an elliptic curve E over a number field K, we prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. As an example, we briefly discuss E=X_1(11) over Q, and how the result relates to Iwasawa theory.

The fine Tate–Shafarevich group

AbstractWithin the Tate–Shafarevich group of an elliptic curve E defined over a number field K, there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p. This group appears naturally in the Iwasawa theory for E. We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.

Computations in non-commutative Iwasawa theory

We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension $Q(\mu_p^\infty,\sqrt[p^\infty]{m})/Q$. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory.

Iwasawa theory of the fine Selmer group

The fine Selmer group of an elliptic curve E E over a number field K K is obtained as a subgroup of the usual Selmer group by imposing stronger conditions at places above p p . We prove a formula for the Euler-characteristic of the fine Selmer group over a Z p \mathbb {Z}_p -extension and use it to compute explicit examples.

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.

Explicit Reciprocity Law for Lubin–Tate Formal Groups

In this article, using Fontaine’s ΦΓ–module theory, we give a new proof of Coleman’s explicit reciprocity law, which generalizes that of Artin–Hasse, Iwasawa andWiles, by giving a complete formula for the norm residue symbol on Lubin–Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.

Analogue of Kida's formula for certain strongly admissible extensions

We study the Iwasawa theory of elliptic curves over certain infinite (non-commutative) p-adic Galois–Lie extensions. In particular, we consider the analogue of the classical Iwasawa λ-invariant and Kida's formula for the dual Selmer group.

On the equivariant main conjecture of Iwasawa theory

Recently, D. Burns and C. Greither (Invent. Math., 2003) deduced an equivariant version of the main conjecture for abelian number fields. This was the key to their proof of the equivariant Tamagawa number conjecture. A. Huber and G. Kings (Duke Math. J., 2003) also use a variant of the Iwasawa main conjecture to prove the Tamagawa number conjecture for Dirichlet motives. We use the result of the second pair of authors and the Theorem of Ferrero-Washington to reprove the equivariant main conjecture in a slightly more general form. The main idea of the proof is essentially the same as in the paper of D. Burns and C. Greither, but we can replace complicated considerations of Iwasawa $mu$-invariants by a considerably simpler argument.