Por Angus Macintyre (Queen Mary, University of London) (Emeritus).

Abstract: To each number field K there is attached a locally compact ring A_K, the ring of adeles over K. This ring is built from the completions of K at equivalence classes of absolute values. These can be either p-adic or real, or complex. Harmonic analysis on the adeles is a fundamental technique in number theory (since the famous thesis of John Tate) .

We analyze definable sets in adele rings, and get definitive information on their topological structures, attempting to relate their measures to zeta functions. We use classical work by Feferman and Vaught to get a ring-theoretic quantifier elimination for adele rings.

We give elementary invariants of adele rings, and prove decidability results. This is connected to number-theoretic work on how much of K one can retrieve from the ring A_K.

(Joint with J. Derakhshan)