Arithmetic of Diagonal Hypersurfaces over Finite Fields by Fernando Q. Gouvêa

By Fernando Q. Gouvêa

There's now a wide physique of thought pertaining to algebraic forms over finite fields, and plenty of conjectures during this sector are of significant curiosity to researchers in quantity thought and algebraic geometry. This booklet bargains with the mathematics of diagonal hypersurfaces over finite fields, with certain concentrate on the Tate conjecture and the Lichtenbaum-Milne formulation for the significant price of the L-function. It combines theoretical and numerical paintings, and contains tables of Picard numbers. even if this ebook is geared toward specialists, the authors have integrated a few heritage fabric to assist nonspecialists achieve entry to the consequences.

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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.

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