We express integrals of definable functions over definable sets uniformly for non‐Archimedean local fields, extending results of Pas. We apply this to Chevalley groups, in particular, proving that zeta functions counting conjugacy classes in congruence quotients of such groups depend only on the size of the residue field, for sufficiently large residue characteristic. In particular, the number of conjugacy classes in a congruence quotient depends only on the size of the residue field. The same holds for zeta functions counting dimensions of Hecke modules of intertwining operators associated to induced representations of such quotients.

We give a definition, in the ring language, of Z p inside Q p and of F p [[t]] inside F p ((t)), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z p inside Q p uniformly for all p. For any fixed finite extension of Q p , we give an existential formula and a universal formula in the ring language which define the valuation ring.

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Abstract. We give foundational results for the model theory of A f in K , the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab-Kuhlmann formalism.