Zeta Functions II: p-adic Analytic Groups

Abstract

As we saw in the last chapter, the arithmetic of subgroup growth in a finitely generated nilpotent group T can be studied ‘locally’: on the one hand, the sequence (a _n (Γ)) is determined in a simple way by the numbers a_pj (Γ) (for all prime-powers p ^j); on the other hand, for each fixed prime p the sequence (a _pJ (Γ)) satisfies a linear recurrence relation: in other words, the local zeta function

$${\zeta _{\Gamma ,p}}(S) = \sum\limits_{j = 0}^\infty {\frac{{{a_{{p^j}}}(\Gamma )}}{{{P^s}}}} $$

(1)

is a rational function in the variable p^-s. The first, ‘global’, property is a special feature of nilpotent (or more generally pronilpotent) groups. The second, ‘local’, one, however, holds in much greater generality. In this chapter we give a brief account of the results, some of the ideas behind them, and some remarkable applications to the enumeration and classification of finite p-groups. For more information, see the detailed survey article [du Sautoy & Segal 2000] and the original papers [du Sautoy 1993], [du Sautoy 2000].