Completion of restricted Lie algebras

We study pro-‘finite dimensional finite exponent’ completions of restricted Lie algebras over finite fields of characteristicp. These compact Hausdorff topological restricted Lie algebras, called pro- restricted Lie algebras, are the restricted Lie-theoretic analogues of pro-p groups. A structure theory for pro- restricted Lie algebras with finite rank is developed. In particular, the centre of such a Lie algebra is shown to be open. As an application we examinep-adic analytic pro-p groups in terms of their associated pro- restricted Lie algebras. Supported by NSERC of Canada.     

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 some Λ-analytic pro-p groups

This paper is devoted to the first steps towards a systematic study of pro-p groups which are analytic over a commutative Noetherian local pro-p ring Λ, e.g. Λ= . We restrict our attention to Λ-standard groups, which are pro-p groups arising from a formal group defined over Λ. Under some additional assumptions we show that these groups are of ‘intermediate growth’ in various senses, strictly betweenp-adic analytic pro-p groups and free pro-p groups. This suggests a refinement of Lazard's theory which stresses the dichotomy betweenp-adic analytic pro-p groups and all the others. In the course of the discussion we answer a question posed in [LM1], and settle two conjectures from [Bo].     

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.     

Kirillov's Orbit Method for p-Groups and Pro-p Groups

In this text, we study Kirillov's orbit method in the context of Lazard's p-saturable groups when p is an odd prime. Using this approach we prove that the orbit method works in the following cases: torsion free p-adic analytic pro-p groups of dimension smaller than p, pro-p Sylow subgroups of classical groups over ? _p of small dimension and for certain families of finite p-groups.     

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact p-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a “generalized Robba ring” for uniform pro-p groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a selfdual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.     

Completed cohomology of Shimura curves and a p-adic Jacquet–Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet–Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.     

On the μ-invariant of two-variable primitive p-adic L-functions

Using the idea of Sinnott,Gillard and Schneps,we prove theμ-invariant is zero for the two-variable primitive p-adic L-function constructed by Kang(2012),which arises naturally in the study of Iwasawa theory for an elliptic curve with complex multiplication(CM).     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Weak Leopoldt's conjecture for Hecke characters of imaginary quadratic fields

We give a proof of the weak Leopoldt's conjecture à la Perrin-Riou, under some technical condition, for the p-adic realizations of the motive associated to Hecke characters over an imaginary quadratic field K of class number 1, where p is a prime >3 and where the CM elliptic curve associated to the Hecke character has good reduction at the primes above p in K. This proof makes use of the 2-variable Iwasawa main conjecture proved by Rubin. Thus we prove the Jannsen conjecture for the above p-adic realizations for almost all Tate twists.     

Iwasawa invariants and Kummer conguruences

Let f(x, χ) be the Iwasawa power series for the p-adic L-function L_p(s, χ), where χ is an even nonprincipal character with conductor not divisible by p² (or by 8, when p = 2). The divisibility by p of the first p coefficients of f(x, χ) is characterized by Kummer type congruences of generalized Bernoulli numbers. Applications to Iwasawa invariants and class numbers of imaginary Abelian fields are discussed.     

On the Lie theory of <Emphasis Type="Italic">p</Emphasis>-adic analytic groups

The aim of this paper is to fill a small, but fundamental, gap in the theory of p-adic analytic groups. We illustrate by example that the now standard notion of a uniformly powerful pro-p group is more restrictive than Lazards concept of a saturable pro-p group. For instance, the Sylow-pro-p subgroups of many classical groups are saturable, but need not be uniformly powerful. Extending work of Ilani, we obtain a correspondence between subgroups and Lie sublattices of saturable pro-p groups. This leads to applications, for instance, in the subject of subgroup growth.Mathematics Subject Classification (2000): 22E20     

A sufficient condition for Leopoldt's conjecture

In this paper, by using an analogue of theorems of Iwasawa (Kenkichi Iwasawa Collected Papers, vol. 2, Springer, Berlin, 2001, pp. 862-870) we give a sufficient condition for Leopoldt's conjecture (J. Reine Angew. Math. 209 (1962) 54) on the non-vanishing of the p-adic regulator of an algebraic number field. Using this sufficient condition we are able to prove Leopoldt's conjecture for several non-Galois extensions over the rational number field Q.     

SK_1 and Lie algebras

We investigate the vanishing of the group $SK_1(\Lambda (G))$ for the Iwasawa algebra $\Lambda (G)$ of a pro- $p$ $p$ -adic Lie group $G$ (with $p \ne 2$ ). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.     

Adjoint selmer groups as Iwasawa modules

We fix a primep. In this paper, starting from a given Galois representation ? having values inp-adic points of a classical groupG, we study the adjoint action of ? on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(?) of ?. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(?))_/L for each number fieldL. We scrutinize the behavior of Sel(Ad(?))_{/E ∞} as an Iwasawa module for a fixed ? _p -extensionE _∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ?. WhenG=GL(2), ? is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.     

Virtually free pro-<Emphasis Type="Italic">p</Emphasis> products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).     

Multifractals,Mumford curves and eternal inflation

We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes related to quantum statistical mechanical systems on Cuntz-Krieger algebras. We extend the eternal inflation model from the Bruhat-Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.