Compact P-adic Analytic Groups Research Articles

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 \({\mathbb {G}}_n\).

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.

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.

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.

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.

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.

Zeta function of representations of compact 𝑝-adic analytic groups

Let G G be an FAb compact p p -adic analytic group and suppose that p > 2 p>2 or p = 2 p=2 and G G is uniform. We prove that there are natural numbers n 1 , … , n k n_1, \ldots , n_k and functions f 1 ( p − s ) , … , f k ( p − s ) f_1(p^{-s}),\ldots , f_k(p^{-s}) rational in p − s p^{-s} such that \[ ζ G ( s ) = ∑ λ ∈ Irr ⁡ ( G ) λ ( 1 ) − s = ∑ i = 1 k n i − s f i ( p − s ) . \zeta ^G(s)=\sum _{\lambda \in \operatorname {Irr}(G)} \lambda (1) ^{-s}=\sum _{i=1}^kn_i^{-s}f_i(p^{-s}). \]