Definable groups and compact p -adic Lie groups

Authors

Alf Onshuus, Anthony L. Pillay

Type: Article

Publication Date: 2008-05-19

Citations: 35

DOI: https://doi.org/10.1112/jlms/jdn018

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Abstract

We formulate p-adic analogues of the o-minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162]; that is, we formulate versions that are appropriate for groups G definable in (saturated) P-minimal fields. We then restrict our attention to saturated models K of Th(ℚp) and Th(ℚp, an), record some elementary observations when G is defined over the standard model ℚp, and then make a detailed analysis of the case where G = E(K) for E an elliptic curve over K. Essentially, our P-minimal conjectures hold in these contexts and, moreover, our case study of elliptic curves yields counterexamples to a more naive direct translation of the o-minimal conjectures.

Locations

Journal of the London Mathematical Society
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2017-01-01

Davide Penazzi , Anand Pillay , Ningyuan Yao

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its … We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields. We consider the additive and multiplicative groups of Q_p and Z_p, the group of upper triangular invertible 2\times 2 matrices, SL(2,Z_p), and, our main focus, SL(2,Q_p). In all cases we identify f-generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the ``Ellis group" of SL(2,Q_p)$ is the profinite completion of Z, yielding a counterexample to Newelski's conjecture with new features: G = G^{00} = G^{000} but the Ellis group is infinite. A final section deals with the action of SL(2,Q_p) on the type-space of the projective line over Q_p.

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Definably Topological Dynamics of $p$-Adic Algebraic Groups

2021-01-01

Jiaqi Bao , Ningyuan Yao

We study the $p$-adic algebraic groups $G$ from the definable topological-dynamical point of view. We consider the case that $M$ is an arbitrary $p$-adic closed field and $G$ an algebraic … We study the $p$-adic algebraic groups $G$ from the definable topological-dynamical point of view. We consider the case that $M$ is an arbitrary $p$-adic closed field and $G$ an algebraic group over ${\mathbb Q}_p$ admitting an Iwasawa decompostion $G=KB$, where $K$ is open and definably compact over ${\mathbb Q}_p$, and $B$ is a borel subgroup of $G$ over ${\mathbb Q}_p$. Our main result is an explicit description of the minimal subflow and Ellis Group of the universal definable $G(M)$-flow $S_G(M^{\text{ext}})$. We prove that the Ellis group of $S_G(M^{\text{ext}})$ is isomorphic to the Ellis group of $S_B(M^{\text{ext}})$, which is $B/B^0$. As applications, we conclude that the Ellis groups corresponding to $\text{GL}(n,M)$ and $\text{SL}(n,M)$ are isomorphic to $(\hat {\mathbb Z} \times {\mathbb Z}_p^*)^n$ and $(\hat {\mathbb Z} \times {\mathbb Z}_p^*)^{n-1}$ respectively, generalizing the main result of Penazzi, Pillay, and Yao in Some model theory and topological dynamics of $p$-adic algebraic groups, Fundamenta Mathematicae, 247 (2019), pp. 191--216.

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Definable Topological Dynamics for Trigonalizable Algebraic Groups over Qp

2019-01-01

Ningyuan Yao

We study the flow (G(Qp); SG(Qp)) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost … We study the flow (G(Qp); SG(Qp)) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f-generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We will give a description of of f-generic types of trigonalizable algebraic groups, and prove that every f-generic type is almost periodic.

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Finding generically stable measures

2012-01-20

Pierre Simon

Abstract This work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures … Abstract This work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.

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ON NON-COMPACT p-ADIC DEFINABLE GROUPS

2021-11-22

Will Johnson , Ningyuan Yao

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A note on fsg groups in p-adically closed fields

2021-08-13

Will Johnson

Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The … Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The case $M = \mathbb{Q}_p$ was previously proved by Onshuus and Pillay.

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Generically stable and smooth measures in NIP theories

2012-12-13

Ehud Hrushovski , Anand Pillay , Pierre Simon

We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by … We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.

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Keisler measures in the wild

2023-06-26

Gabriel Conant , Kyle Gannon , James Hanson

We investigate Keisler measures in arbitrary theories.Our initial focus is on Borel definability.We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under … We investigate Keisler measures in arbitrary theories.Our initial focus is on Borel definability.We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and satisfy associativity.However, we also demonstrate failures of both properties over uncountable parameter sets.In particular, we show that the Morley product of Borel definable types need not be Borel definable (correcting an erroneous result from the literature).We then study various notions of generic stability for Keisler measures and generalize several results from the NIP setting to arbitrary theories.We also prove some positive results for the class of frequency interpretation measures in arbitrary theories, namely, that such measures are closed under convex combinations and commute with all Borel definable measures.Finally, we construct the first example of a complete type which is definable and finitely satisfiable in a small model, but not finitely approximated over any small model.

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Open Subgroups of p-Adic Algebraic Groups

2025-01-01

Anand Pillay , Ningyuan Yao

On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.

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References (13)

The Arithmetic of Elliptic Curves

1986-01-01

Joseph H. Silverman

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Advanced Topics in the Arithmetic of Elliptic Curves

1994-01-01

Joseph H. Silverman

Analytic Pro-P Groups

1999-10-28

J. D. Dixon , M. P. F. Du Sautoy , A. Mann +1 more

Prelude Part I. Pro-p Groups: 1. Profinite groups and pro-p groups 2. Powerful p-groups 3. Pro-p groups of finite rank 4. Uniformly powerful groups 5. Automorphism groups Interlude A. Fascicule … Prelude Part I. Pro-p Groups: 1. Profinite groups and pro-p groups 2. Powerful p-groups 3. Pro-p groups of finite rank 4. Uniformly powerful groups 5. Automorphism groups Interlude A. Fascicule de resultats: pro-p groups of finite rank Part II. Analytic Groups: 6. Normed algebras 7. The group algebra Interlude B. Linearity criteria 8. P-adic analytic groups Interlude C. Finitely generated groups, p-adic analytic groups and Poincare series 9. Lie theory Part III. Further Topics: 10. Pro-p groups of finite co-class 11. Dimension subgroup methods 12. Some graded algebras Interlude D. The Golod Shafarevic inequality Interlude E. Groups of sub-exponential growth 13. Analytic groups over pro-p rings.

The Structure of Compact Groups

2006-08-22

Karl H. Hofmann , Sidney A. Morris

Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose … Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. However, the thrust of book points in the direction of the structure theory of infinite dimensional, not necessarily commutative compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups. The first edition of 1998 was well received by reviewers and has been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and some minor inaccuracies of content; it has been edited and improved in various sections. New material has been added in order to reflect ongoing research. In the process of revising the original edition, the integrity of the original section numbering was carefully respected so that citations of material from the first edition remains perfectly viable to the users of this edition.

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01

Ya’acov Peterzil , Charles Steinhorn

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

On definable subsets of p-adic fields

1976-09-01

Angus Macintyre

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically … The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

On fields definable inQ p

1989-03-01

Anand Pillay

An additive measure in o-minimal expansions of fields

2004-12-01

Alessandro Berarducci

Journal Article An additive measure in o-minimal expansions of fields Get access Alessandro Berarducci, Alessandro Berarducci Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero … Journal Article An additive measure in o-minimal expansions of fields Get access Alessandro Berarducci, Alessandro Berarducci Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero Margarita Otero Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 55, Issue 4, December 2004, Pages 411–419, https://doi.org/10.1093/qmath/hah010 Published: 01 December 2004

An application of model theory to real and p-adic algebraic groups

1989-10-01

Anand Pillay

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22

Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

p-adic and Real Subanalytic Sets

1988-07-01

Jan Denef , Lou van den Dries

Open Subgroups of GL 2 (Q p ).

1989-01-01

Ali Nesin , Anand Pillay

Definable groups and compact <i>p</i> -adic Lie groups

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