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A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups. (English) Zbl 0998.20033

In their landmark paper [Invent. Math. 93, No. 1, 185-223 (1988; Zbl 0651.20040)] F. J. Grunewald, D. Segal and G. C. Smith introduced the notion of the zeta function \[ \zeta^\leq_G(s)=\sum_{H\leq G}|G:H|^{-s}=\sum_{n=1}^\infty a^\leq_n(G)\cdot n^{-s} \] for a group G. Here \(a^\leq_n(G)\) stands for the number of subgroups of index \(n\) in \(G\). (One can work in a similar way with normal subgroups.) Based on the available examples, and the analogy with the Dedekind zeta function of a number field, they were led to conjecture that if \(G\) is a finitely generated nilpotent group, then there exist finitely many rational functions \(W_1(X,Y),\dots,W_r(x,y)\in\mathbb{Q}(X,Y)\) such that for each prime \(p\) there is an \(i\) such that \(\zeta^\leq_{G,p}(s)=W_i(p,p^{-s})\). Here \(\zeta^\leq_{G,p}(s)=\sum_{n=1}^\infty a^\leq_{p^n}(G)\cdot p^{-ns}\) is the local zeta function. If this holds for a group \(G\), one says that the local zeta functions are finitely uniform. If one rational function suffices (that is, one can take \(r=1\) above), one says that the local zeta functions are uniform.
Recent work of M. du Sautoy and F. Grunewald [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 5, 351-356 (1999; Zbl 1062.11503), Ann. Math. (2) (to appear)] shows that the behaviour of the local factors as one varies over the prime is not related to the behaviour of primes in number fields, as originally suspected, but rather to the number of points modulo \(p\) of a variety. This number is known to vary wildly with \(p\), so that it does not lend itself to a finitely uniform description. For instance, the number of points modulo \(p\) on the elliptic curve given by \(y^2-x^3+x=0\) is \(p\) if \(p\equiv 3\pmod 4\); when \(p\equiv 1\pmod 4\), though, the number is \(p-2a\), where \(p=a^2+b^2\), and \(a+ib\equiv 1\pmod{2+2i}\).
The purpose of the paper under review is to exhibit a nilpotent group \(G\) whose zeta function depends on the behaviour modulo \(p\) of the elliptic curve above, and thus does not lend itself to a finitely uniform description. This is a group of nilpotency class two and Hirsch length \(9\). The equation describing the elliptic curve is cleverly embedded in the commutator pattern. It is possible that this method can be extended to encode an arbitrary variety. Although it is well-known that nilpotent groups of class two can be quite wild taken as a whole, the new phenomena occurring within an individual group, brought to light by this paper, are striking.
Reviewer: A.Caranti (Povo)

MSC:

20F18 Nilpotent groups
11M41 Other Dirichlet series and zeta functions
14H52 Elliptic curves
20E07 Subgroup theorems; subgroup growth
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