Perhaps the most successful attempt at analyzing first order theory of p-adic fields is through the use of RV language (aka. leading term structure). In this, quantifier elimination on the field sort can be removed leaving us with only quantifiers on the RV sort, and the process can be described very explicitly. From this, many classical results such as Mcintyre's quantifier elimination of finitely ramified p-adic field in mixed characteristic can be deduced.
However, these RV groups can change rather wildly dependent on the field, so without adding in additional assumption that would fix them down, it seemed impossible to analyze. So I guess I have 2 questions:
What are some nontrivial first-order property known to be true for almost all finite extensions of $\mathbb{Q}_{p}$? (for example, for finite field that would be Weil conjecture)
What are some of the recent attempts at analyzing the uniform first order theory of finite extension of $\mathbb{Q}_{p}$? For example, perhaps working toward an uniform quantifier elimination, or a proof that such attempt is undecidable?
Thank you very much for your help.
