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Definable groups and compact p ‐adic Lie groups
Author(s) -
Onshuus A.,
Pillay A.
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn018
Subject(s) - counterexample , mathematics , elliptic curve , group (periodic table) , combinatorics , pure mathematics , physics , quantum mechanics
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(ℚ p ) and Th(ℚ p, an ), record some elementary observations when G is defined over the standard model ℚ p , 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.