By Korchmaros G.

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5). This work is based on the classiﬁcation of such groups by Grunewald and Segal [31]. For centres of rank 3, the geometry of the associated Pfaﬃan hypersurface comes into play. Provided the singularities of this hypersurface are in some sense not too severe, Voll gives a formula for the local normal zeta function of L 22 2 Nilpotent Groups: Explicit Examples depending on the number of points on the Pfaﬃan hypersurface. 6). A more general approach is used by Voll in [61], where he considers the case where the Pfaﬃan hypersurface has no lines.

Hironaka’s proof of resolution of singularities of any singular variety deﬁned over a ﬁeld of characteristic 0 has been reﬁned by Villamayor, Encinas, Bierstone and Milman, and Hauser amongst others to produce an explicit constructive procedure. In particular, Bodn´ ar and Schicho have implemented a computer program to calculate resolutions. We refer the reader wanting to know more to Hauser’s accessible article on resolution [34] and its comprehensive bibliography. However, we shall not use resolution of singularities, for a number of reasons.

G4 ,p (s) = ζZ4 ,p (s)ζp (3s − 6)ζp (5s − 10)ζp (7s − 12)WG4 (p, p−s ) , where WG4 (X, Y ) = 1 + X 4 Y 3 + X 5 Y 3 + X 8 Y 5 + X 9 Y 5 + X 13 Y 8 , and ζG≤4 ,p (s) = ζZ4 ,p (s)ζp (2s − 5)ζp (2s − 6)ζp (2s − 7)ζp (3s − 10)ζp (4s − 12) × WG≤4 (p, p−s ) where WG≤4 (X, Y ) is 1 + X 4Y 2 + X 5Y 2 + X 6Y 2 − X 5Y 3 − X 6Y 3 − X 7Y 3 + X 8Y 3 + X 9Y 3 − X 9 Y 4 − X 10 Y 4 − X 11 Y 4 − X 14 Y 6 − X 15 Y 6 − X 16 Y 6 + X 16 Y 7 + X 17 Y 7 − X 18 Y 7 − X 19 Y 7 − X 20 Y 7 + X 19 Y 8 + X 20 Y 8 + X 21 Y 8 + X 25 Y 10 .