By Korchmaros G.
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Extra info for 2-transitive abstract ovals of odd order
5). This work is based on the classiﬁcation of such groups by Grunewald and Segal . 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 , 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  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 .