Definable groups and compact p ‐adic Lie groups

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.