"Model theory of local fields and counting problems in Chevalley groups"

This is joint with with Mark Berman, Uri Onn, and Pirita Paajanen.

Let K be a local field with valuation ring O and residue field of size q, and G a Chevalley group. We study counting problems associated with the group G(O). Such counting problems are encoded in certain zeta functions defined as Poincare series in q^{-s}. It turns out that these zeta functions are bounded sums of rational functions and depend only on q for all local fields of sufficiently large residue characteristic. We apply this to zeta functions counting conjugacy classes or dimensions of Hecke modules of interwining operators in congruence quotients of G(O). To prove this we use model-theoretic cell decomposition and quantifier-elimination to get a theorem on the values of 'definable' integrals over local fields as the local field varies.