Non-commutative Iwasawa Theory Research Articles

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Abstract The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory are developed.

Arithmetic statistics and noncommutative Iwasawa theory

Let p be an odd prime. Associated to a pair (E, \mathcal{F}_\infty) consisting of a rational elliptic curve E and a p -adic Lie extension \mathcal{F}_\infty of \mathbb{Q} , is the p -primary Selmer group \mathrm{Sel}_{p^{\infty}}(E/\mathcal{F}_\infty) of E over \mathcal{F}_\infty . In this paper, we study the arithmetic statistics for the algebraic structure of this Selmer group. The results provide insights into the asymptotics for the growth of Mordell-Weil ranks of elliptic curves in noncommutative towers.

The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture

Let K denote a number field containing a primitive p -th root of unity; if p=2 , then we assume K to be totally imaginary. If K_\infty/K is a \mathbb{Z}_p -extension such that no prime above p splits completely in K_\infty/K , then the vanishing of Iwasawa's invariant \mu(K_\infty/K) implies that the weak Leopoldt Conjecture holds for K_\infty/K . This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible p -adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible p -adic Lie extensions.

Some remarks on Kida’s formula when $$\mu \ne 0$$

The Kida's formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an appropriate $\mu=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $\mu\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.

Control theorem and functional equation of Selmer groups over p-adic Lie extensions

Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even in the case where the Selmer group is associated to an elliptic curve. We also discuss the functional equation for the analytic $p$-adic $L$-functions and check the compatibility with the functional equation of the dual Selmer groups.

On derivatives of p-adic L-series at s = 0

Abstract We use techniques of non-commutative Iwasawa theory to investigate the values at zero of higher derivatives of p-adic Artin L-series.

$$\hbox {K}_{1}$$ K 1 -congruences for three-dimensional Lie groups

We completely describe $$\hbox {K}_{1}({\mathbb {Z}}_p[\![{\mathcal {G}}_{\infty }]\!])$$ and its localisations by using an infinite family of p-adic congruences, where $${\mathcal {G}}_{\infty }$$ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when $$\hbox {dim}({\mathcal {G}}_{\infty })=2$$ , and of the first named author and Lloyd Peters when $${\mathcal {G}}_{\infty } \cong {\mathbb {Z}}_p^{\times }\ltimes {\mathbb {Z}}_p^d$$ with a scalar action of $${\mathbb {Z}}_p^{\times }$$ . The method exploits the classification of 3-dimensional p-adic Lie groups due to Gonzalez-Sanchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory.

Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

AbstractLet$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

On p-adic L-series, p-adic cohomology and class field theory

Abstract We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic ℤ p {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

Non-abelian p-adic L-functions and Eisenstein series of unitary groups – The CM method

In this work we prove various cases of the so-called “torsion congruences” between abelian p-adic L-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of n variables and we obtain more explicit results in the special cases of n = 1 and n = 2. In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”.

P-adic L-functions over the false Tate curve extensions

AbstractLet f be a primitive modular form of CM type of weight k and level Γ0(N). Let p be an odd prime which does not divide N, and for which f is ordinary. Our aim is to p-adically interpolate suitably normalized versions of the critical values L(f, ρχ,n), where n=1,2,. . .,k − 1, ρ is a fixed self-dual Artin representation of M∞ defined by (1.1) below, and χ runs over the irreducible Artin representations of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, if k ≥ 4, we will show that there are only finitely many χ such that L(f, ρχ,k/2)=0, generalizing a result of David Rohrlich. Also, we conditionally establish a congruence predicted by non-commutative Iwasawa theory and give numerical evidence for it.

ON μ-INVARIANTS OF SELMER GROUPS OF SOME CM ELLIPTIC CURVES

We show that the Selmer groups defined over the cyclotomic ℤ3-extension of the field of 3-torsion points of the CM elliptic curves 256a1, 256a2, 256d1, 256d2, 121b1, 121b2 have μ-invariant equal to zero, verifying a conjecture in non-commutative Iwasawa theory.

On main conjectures in non-commutative Iwasawa theory and related conjectures

Abstract The main conjecture of non-commutative Iwasawa theory is shown to be equivalent, modulo an appropriate torsion hypothesis, to a family of main conjectures over cyclotomic ℤ p -extensions together with suitable compatibility relations between the associated abelian p-adic L-functions. This result combines with techniques of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures for (non-abelian) compact p-adic Lie extensions of arbitrarily large rank from main conjectures in the classical sense. As a first application of this approach we combine it with a recent result of Ritter and Weiss to prove, modulo an appropriate vanishing hypothesis on μ-invariants, the main conjecture of non-commutative Iwasawa theory for Tate motives over compact p-adic Lie extensions of totally real number fields of arbitrarily large rank. We then also deduce several interesting consequences of this result concerning a range of related conjectures including special cases of the equivariant Tamagawa number conjecture.

Non-commutative Iwasawa theory for modular forms

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k > 2 over the Galois extension of ℚ obtained by adjoining to ℚ all p-power roots of unity, and all p-power roots of a fixed integer m > 1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agree perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.

On a Localisation Sequence for the K-Theory of Skew Power Series Rings

AbstractLet B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B = ℤp[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ≅ H ⋊ ℤp, this set S is Venjakob's canonical Ore set. In particular, our result implies thatis split exact for each n ≥ 0. We also prove the corresponding result for the localisation of ℤp[[G]][$\frac{1}{p}$] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.

IWASAWA THEORY FOR ONE-PARAMETER FAMILIES OF MOTIVES

In [A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proc. St. Petersburg Mathematical Society, Vol. 12, American Mathematical Society Translations, Series 2, Vol. 219 (American Mathematical Society, Providence, RI, 2006), pp. 1–85] Fukaya and Kato presented equivariant Tamagawa number conjectures that implied a very general (non-commutative) Iwasawa main conjecture for rather general motives. In this article we apply their methods to the case of one-parameter families of motives to derive a main conjecture for such families. On our way there we get some unconditional results on the variation of the (algebraic) λ- and μ-invariant. We focus on the results dealing with Selmer complexes instead of the more classical notion of Selmer groups. However, where possible we give the connection to the classical notions.

The main conjecture of Iwasawa theory for totally real fields

Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of $\mathcal{G}$ in terms of Iwasawa algebras of Abelian subquotients of $\mathcal{G}$ . We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of $\mathcal{G}$ , which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa’s conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ.

Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction

AbstractThis paper studies the compactp∞-Selmer Iwasawa moduleX(E/F∞) of an elliptic curveEover a False Tate curve extensionF∞, whereEis defined over ℚ, having multiplicative reduction at the odd primep. We investigate thep∞-Selmer rank ofEover intermediate fields and give the best lower bound of its growth under certain parity assumption onX(E/F∞), assuming this Iwasawa module satisfies theH(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.

On prime ideals of noetherian skew power series rings

We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern “Cutting Down” and “Lying Over.” In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ − id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings.

NON-ABELIAN CONGRUENCES BETWEEN SPECIAL VALUES OF L-FUNCTIONS OF ELLIPTIC CURVES: THE CM CASE

In this work we prove congruences between special values of L-functions of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analog for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert–Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of L-values.