DEFINABLE GROUPS AND COMPACT p-ADIC LIE GROUPS ALF ONSHUUS AND ANAND PILLAY Abstract. We formulate a version of the o-minimal group con- jectures from [11], which is appropriate for groups G definable in a (saturated) p-adically closed field K, We discuss the conjectures in two cases, when G is defined over Q p and when G is of the form E(K) for E an elliptic curve over K. 1. Introduction In [11] questions were raised concerning recovering a compact Lie group from a definable group G in saturated o-minimal structures, by quotienting by a type-definable subgroup of G of bounded index. Related work was done in [1], [2] and [9]. As Gregory Cherlin suggested to us, it is rather natural to consider p-adic analogues of the questions. This is what we discuss in the present paper. The right level of model- theoretic generality is probably that of so-called P -minimal expansions of p-adically closed fields. However we restrict ourselves here to p- adically closed fields. In fact for notational convenience, we look at groups definable in a saturated elementary extension (K, +, ·) of the p- adic field Q p . So one could view this as the study of uniformly definable families of groups in Q p . Before stating the conjectures and results of this paper, we give some definitions, state some elementary facts, and recall some facts about p-adic analytic groups. We refer to [5] for background on compact groups and profinite groups, and to [4] for background on p-adic ana- lytic groups. To begin with let ¯ M be a saturated model (of cardinality κ> |T | say where κ is inaccessible) of an arbitrary complete theory T in a language L, and let G be a group definable in ¯ M . Let F G denote the family of definable subgroups of G of finite index. Definition 1.1. (i) Suppose that the family F G of definable subgroups of G of finite index has cardinality <κ. Then we define G 0 to be ∩F , and we also say that “G 0 exists”. 1