Zeta functions and counting finite $p$-groups

We announce proofs of a number of theorems concerning finite $p$-groups and nilpotent groups. These include: (1) the number of $p$-groups of class $c$ on $d$ generators of order $p^n$ satisfies a linear recurrence relation in $n$; (2) for fixed $n$ the number of $p$-groups of order $p^n$ as one varies $p$ is given by counting points on certain varieties mod $p$; (3) an asymptotic formula for the number of finite nilpotent groups of order $n$; (4) the periodicity of trees associated to finite $p$-groups of a fixed coclass (Conjecture P of Newman and O'Brien). The second result offers a new approach to Higman's PORC conjecture. The results are established using zeta functions associated to infinite groups and the concept of definable $p$-adic integrals.