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

i.e. jp(α) with
α ∈
1
6
, 1
4
1 1
, , ,
4 3
5
6
, 1
4
1 2
, ,
4 3
and their conjugates ¯α = 1−α, where 1 denotes the unit vector. The values of the independent
Jacobi sums are collected for low p f in Table 1.
p f j p f
1
6
, 1
4
, 1
4
1 , 3
13 −3 + 4 √ 3 + 2 + 6 √ 3 i −3 − 4 √ 3 + 2 − 6 √ 3 i
25 15 − 20i 15 − 20i
37 −5 − 12 √ 3 + 30 − 2 √ 3 i −5 + 12 √ 3 + 30 + 2 √ 3 i
49 −7 − 28 √ 3i −7 + 28 √ 3i
61 35 − 12 √ 3 − 42 + 10 √ 3 i 35 + 12 √ 3 − 42 − 10 √ 3 i
73 15 − 32 √ 3 + 40 + 12 √ 3 i 15 + 32 √ 3 + 40 − 12 √ 3 i
Table 1. Jacobi sums for the K3 surface X 12
2 ⊂ P(2,3,3,4).
Using these results leads to the L-function of the Ω−motive
j p f
5
6
, 1
4
, 1
4
2 , 3
(75)
LΩ(X 12
2 , s) . = 1 − 12 30 20 14 140 60
+ − − + + + · · · (76)
13s 25s 37s 49s 61s 73s Insight into the structure of this L-function can be obtained by noting that the surface X 12
2
can be constructed via the twist map [43, 44] (see also [45, 46]) by considering
defined by
Φ : P(2,1,1) × P(3,1,2) −→ P(3,3,2,4) (77)
((x0, x1, x2), (y0, y1, y2)) ↦→
y 1/3
0 x1, y 2/3
0 x2, x 1/2
0 y1, x 1/2
0 y2
, (78)
which on the product E 4 − × E 6 leads to the K3 surface of degree twelve X 12
2 .
The coefficients of L(E 4 − , s) = L(E4 , s) =
n an(E 4 )n −s and L(E 6 , s) =
n bn(E 6 )n −s of
the Hasse-Weil L-functions of the elliptic curves E 4 and E 6 , respectively, can be obtained by
expanding the results of Theorem 6. Multiplying the coefficients ap(E 4 ) and bp(E 6 ) leads
For low primes the results are collected in Table 2.
ap(E 4 )bp(E 6 ) = cp(X 12
2 ). (79)
25