arXiv:0812.4450v1 [hep-th] 23 Dec 2008

For any degree vector n = (n0, ..., ns+1) and for any prime p define the numbers di = (ni, p−1)
and the set
(α0, ..., αs+1) ∈ Q s+2 | 0 < αi < 1, diαi = 0 (mod 1),
αi = 0 (mod 1) . (58)
A p,n
s =
Theorem 5. The number of solutions of the smooth projective variety
s+1
Xs = (z0 : z1 : · · · : zs+1) ∈ Ps+1 | biz ni
i = 0 ⊂ Ps+1
over the finite field Fp is given by
where
i=0
Np(Xs) = 1 + p + p 2 + · · · + p s +
jp(α) = 1
p − 1
P s p (t) = (1 − ps/2
|s|
t)
u i ∈IFq
u0+···+us=0
α∈A n s
α∈A p,n
s
i
(59)
jp(α) ¯χαi (ai), (60)
χα0(u0) · · ·χαs(us). (61)
With these Jacobi sums jq(α) one defines the polynomials
1 − (−1) s jpf(α)
¯χαi (bi )t f
and the associated L-function
L (j) (X, s) =
Here |s| = 1 if s is even and |s| = 0 if s is odd.
p
1
P j p(p −s )
i
1/f
(62)
. (63)
A slight modification of this result is useful even in the case of smooth weighted projective
varieties because it can be used to compute the factor of the zeta function coming from the
invariant part of the cohomology, when viewing these spaces as quotient varieties of projective
spaces.
The Jacobi-sum formulation allows to write the L-function of the Ω−motive MΩ of weighted
k0 ks+1
Fermat hypersurfaces in a more explicit way. Define the vector αΩ = , ..., corre-
d d
sponding to the holomorphic s−form, and denote its Galois orbit by OΩ ⊂ An s . Then
LΩ(X, s) = s
1 − (−1) jpf(α)p −fs−1/f . (64)
p
α∈OΩ
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