P-adic Analytic Groups Research Articles

Rationality of twist representation zeta functions of compact 𝑝-adic analytic groups

We prove that for any twist rigid compact p p -adic analytic group G G , its twist representation zeta function is a finite sum of terms n i − s f i ( p − s ) n_{i}^{-s}f_{i}(p^{-s}) , where n i n_{i} are natural numbers and f i ( t ) ∈ Q ( t ) f_{i}(t)\in \mathbb {Q}(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G G is moreover a pro- p p group, we prove that its twist representation zeta function is rational in p − s p^{-s} . To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

Abstract Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

Maximal prime homomorphic images of mod-p Iwasawa algebras

AbstractLet k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

Index-stable compact $${\varvec{p}}$$-adic analytic groups

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let p be a prime, and let G be a compact p-adic analytic group with associated $$\mathbb {Q}_p$$ -Lie algebra $$\mathcal {L}(G)$$ . We prove that G is index-stable whenever $$\mathcal {L}(G)$$ is semisimple. In particular, a just-infinite compact p-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact p-adic analytic groups, this gives a positive answer to a question of C. Reid. In the appendix, J-P. Serre proves that G is index-stable if and only if the determinant of any automorphism of $$\mathcal {L}(G)$$ has p-adic norm 1.

On the catenarity of virtually nilpotent mod-p Iwasawa algebras

Let p > 2 be a prime, k a finite field of characteristic p, and G a nilpotent-by-finite compact p-adic analytic group. Write kG for the completed group ring of G over k. We show that kG is a catenary ring.

On self-similarity of p-adic analytic pro-p groups of small dimension

Given a torsion-free p-adic analytic pro-p group G with dim(G)<p, we show that the self-similar actions of G on regular rooted trees can be studied through the virtual endomorphisms of the associated Zp-Lie lattice. We explicitly classify 3-dimensional unsolvable Zp-Lie lattices for p odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free p-adic analytic pro-p groups for p⩾5, and to determine which of them admit a faithful self-similar action on a p-ary tree. In particular, we show that no open subgroup of SL11(Δp) admits such an action. On the other hand, we prove that all the open subgroups of SL2△(Zp) admit faithful self-similar actions on regular rooted trees.

Reflexive ideals and completely faithful modules of Iwasawa algebras

Let p≥5 be a prime number. Let G=H×Z, where H is a torsion free compact p-adic analytic group such that its Lie algebra is split semisimple over Qp and Z≅Znp, where n∈N0. We classify all the prime c-ideals of the Iwasawa algebra ΛG. Then we show the following: Let M be a finitely generated torsion ΛG-module, such that it has no non-zero pseudo-null submodules. Let q(M) denote the image of M in the quotient category mod-ΛG/CΛG1 via the quotient functor q, where CΛG1 denotes the Serre-subcategory of pseudo-null ΛG-modules in mod-ΛG. Then q(M) is completely faithful if and only if M is ΛZ-torsion free.

Hausdorff dimensions in p-adic analytic groups

Let G be a finitely generated pro-p group, equipped with the p-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of G. Conversely, it is a long-standing open question whether the finiteness of the Hausdorff spectrum implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for odd primes p, that every countably based pro-p group G with an open subgroup mapping onto 2 copies of the p-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.

The base change in the Atiyah and the Lück approximation conjectures

Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.

On the structure of virtually nilpotent compact p-adic analytic groups

Abstract Let G be a compact p-adic analytic group. We recall the well-understood finite radical Δ + {\Delta^{+}} and FC-centre Δ, and introduce a p-adic analogue of Roseblade’s subgroup nio ⁢ ( G ) {\mathrm{nio}(G)} , the unique largest orbitally sound open normal subgroup of G. Further, when G is nilpotent-by-finite, we introduce the finite-by-(nilpotent p-valuable) radical 𝐅𝐍 p ⁢ ( G ) {\mathbf{FN}_{p}(G)} , an open characteristic subgroup of G contained in nio ⁢ ( G ) {\mathrm{nio}(G)} . By relating the already well-known theory of isolators with Lazard’s notion of p-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble) p-valuable group, and use this to study the conjugation action of nio ⁢ ( G ) {\mathrm{nio}(G)} on 𝐅𝐍 p ⁢ ( G ) {\mathbf{FN}_{p}(G)} . We emerge with a structure theorem for G, 1 ≤ Δ + ≤ Δ ≤ 𝐅𝐍 p ⁢ ( G ) ≤ nio ⁢ ( G ) ≤ G , 1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G, in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group rings kG) of such groups, and will be used in future work to study the prime ideals of these rings.

Locally analytic vectors in representations of locally 𝑝-adic analytic groups

This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any continuous representation of a p-adic analytic group on a locally convex p-adic topological vector space the associated space of locally analytic vectors. Using this functor, and the point of view that its construction suggests, we establish some basic facts about admissible locally analytic representations (as defined by Schneider and Teitelbaum). We also introduce the related notion of essentially admissible locally analytic representations.

The pro-p group of upper unitriangular matrices

We study the pro-p group G whose finite quotients give the prototypical Sylow p-subgroup of the general linear groups over a finite field of prime characteristic p. In this article, we extend the known results on the subgroup structure of G. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski [13–15] studies a free product Cp⁎Cp as a (discrete) subgroup of G and we prove that its closure is selfnormalising of infinite index in the subgroup of 2-periodic elements of G. We also discuss change of rings: field extensions and a variant for the p-adic integers, this latter linking G with some well known p-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of G and show that the Hausdorff spectrum of G is the whole interval [0,1] which is obtained by considering partition subgroups only.

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adèlic profinite groups of type A 2 . This has consequences for the representation zeta functions of arithmetic groups Γ ⊂ H ( k ) , where k is a number field and H is a k-form of SL 3 : assuming that Γ possesses the strong congruence subgroup property, we obtain precise, uniform estimates for the representation growth of Γ. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL 3 ( o ) and SU 3 ( o ) , where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 × 3 matrices in gl 3 ( o ) and gu 3 ( o ) , where o is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form SL 3 ( o ) , SU 3 ( o ) , GL 3 ( o ) , and GU 3 ( o ) , arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of ‘Ennola duality’.

On irreducible representations of compact p-adic analytic groups

We prove that the canonical dimension of a coadmissible representation of a semisimple p-adic Lie group in a p-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit. To do this we establish analogues for p-adically completed enveloping algebras of Bernstein’s inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen’s Lemma about the endomorphism ring of a simple module over an enveloping algebra.

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension. Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

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 ℤ p of small dimension and for certain families of finite p-groups.

CENTRES OF SKEWFIELDS AND COMPLETELY FAITHFUL IWASAWA MODULES

Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebraH of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebraG of G := H × Zp. We also show that a finitely generated torsionG-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

Reflexive ideals in Iwasawa algebras

Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra Ω G of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in Ω G is a unit. We show that these conditions hold in the case when G is an open subgroup of SL 2 ( Z p ) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in Ω G when G is a congruence subgroup of SL 2 ( Z p ) : the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.

K 0 and the dimension filtration for p -torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely generated kG-module with canonical dimension smaller than the dimension of the centralizer, as a p-adic analytic group, of any p-regular element of G, then the Euler characteristic of M is trivial. Writing ℱi for the abelian category consisting of all finitely generated kG-modules of dimension at most i, we provide an upper bound for the rank of the natural map from the Grothendieck group of ℱi to that of ℱd, where d denotes the dimension of G. We show that this upper bound is attained in some special cases, but is not attained in general.