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

9 The K3 fibration hypersurface X 12
3 ⊂ P (2,2,2,3,3)
9.1 The Ω−motive of X 12
2
Consider the weighted Fermat hypersurface of degree twelve in P(2,2,2,3,3) given in eq. (3).
The Galois group of this variety is of order four, leading to an Ω−motive of rank four. The
1 1 1 1 1 1 1 1 3 3
Jacobi sums that parametrize this motive are given by jp , , , , , jp , , , , and
6 6 6 4 4 6 6 6 4 4
their complex conjugates. The computation of these sums for low pf are collected in Table 5.
p f j p f
1
6
, 1
6
, 1
6
13 −3 2 + 15√3 + 1 + 45
2
1 1
1 1 1 3 3
, , j 4 4
pf , , , , 6 6 6 4 4
√
3 3 i −2 − 15√3 − 1 − 45
√
3 i 2
25 75 − 100i 75 + 100i
37 −47 2 + 99√3 + 141 + 33
√
47 3 i − 2
2 − 99√3 − 141 − 33
√
3 i 2
7
49
2 (71 + 39√3i 2 (71 + 39√3i 605 61 2 + 27√3 − 363 − 45
√
605 3 i 2
2 − 27√3 + 363 + 45
√
3 i 2
73 −291 2 + 252√3 − 388 + 189
√
291 3 i − 2
2 − 252√3 + 388 − 189
√
3 i 2
Table 5. Jacobi sums of the Ω−motive of X 12
3 .
Using these results leads to the expansion of the L-series
LΩ(X 12
3
6 150 94 497 1210 582
, s) 1 + − + − − + + · · · (87)
13s 25s 37s 49s 61s 73s The structure of the Ω−motivic L-series of X12 3 can be understood by noting that the threefold
is a K3 fibration with typical fiber X6A 2 given in (69). The interpretation of LΩ(X12 3 , s) in
terms of the fibration is also useful because it makes the complex multiplication structure of
the associated modular form transparent. The threefold can be constructed explicitly as the
quotient of a product of a torus E and a K3 surface
X = E × K3/ι, (88)
where ι is an involution acting on the product. More precisely, the elliptic curve is given by
the weighted
E 4 − = {x2 0 − (x4 1 + x4 2
and the K3 surface is the generic fiber X 6A
2 .
30
= 0)} ⊂ P(2,1,1)
(89)
7