Zeta functions and counting finite p-groups

Abstract

We announce proofs of a number of theorems concerning finite p p -groups and nilpotent groups. These include: (1) the number of p p -groups of class c c on d d generators of order p n p^n satisfies a linear recurrence relation in n n ; (2) for fixed n n the number of p p -groups of order p n p^n as one varies p p is given by counting points on certain varieties mod p p ; (3) an asymptotic formula for the number of finite nilpotent groups of order n n ; (4) the periodicity of trees associated to finite p 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 p -adic integrals.