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**Extra resources for Differentiable Manifolds (1972)**

**Example text**

For each integer i > 1, let ki = F9: and let Ni denote the number of ki-rational points on V = Va(c). Then the zeta-function of V = VV (c) is defined as N Z(V, T) = exp(E2 00 Tz) E Q((T)). i=1 The following properties of Z(V, T) are well known: 1. The zeta-function Z(V, T) is a rational function of the form Z(V, T) = T) (-ln+l qiT) lli-o(1 - where Q(V, T) E 1 + T7G[T] with deg(Q) _ E Q(T) 11 {(m - 1)n+1 + (-1)n+2}. 2. k(co cil an ... cn+j' )i (a) is a twisted Jacobi sum of dimension n and of degree m with absolute value qn/2.

Therefore, all the twisted Fermat motives of type I stemming from this motive are also ordinary and supersingular. (2) Let (m, n) _ (19, n) with n > 1. Let p be a prime such that p - 4 or 5 (mod 19). Take n = 2. Then V[1,4,s,9) is of Hodge-Witt type. Therefore, all the induced twisted Fermat motives of dimension 2 + 2d are of Hodge-Witt type. (Type II) Let (m, n) = (7, n) with n > 1. (1) Let p be a prime such that p - 2 or 4 (mod 7). So f = 3. Let a = (1, 1, 2, 4, 6) E 2(3. Then VA is ordinary.

HodgeWitt, resp. supersingular). 7 If VA is supersingular then j(a) = 6qn/2, where 6 is an m-th root of unity. If m is a prime, m > 3, then in fact j (a) = qn/2. Proof: The first assertion is well known. 10. From the lemma we see that if VA is supersingular then 3(c, a) differs from qn/2 by a factor of a root of unity. This explains the term "strongly supersingular" above. The case of p - 1 (mod m), so that f = 1, is special in many ways. First of all, we have AH(a) = hail for every character a, and this immediately shows Twisted Fermat motives 34 that in this case every motive will be ordinary.