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

Theorem 7. Let MΩ ⊂ H 2 (X d 2) be the irreducible representation of Gal(Q(µd)/Q) associated
to the holomorphic 2−form Ω ∈ H2,0 (Xd 2 ) of the K3 surface Xd 2 , where d = 4, 6A, 6B. Then
the q−series fΩ(Xd 2, q) of the L-functions LΩ(Xd 2, s) are modular forms given by
fΩ(X 4 2 , q) = η6 (q 4 )
fΩ(X 6A
2 , q) = ϑ(q3 )η 2 (q 3 )η 2 (q 9 )
fΩ(X 6B
2 , q) = η 3 (q 2 )η 3 (q 6 ) ⊗ χ3. (70)
These functions are cusp forms of weight three with respect to Γ0(N) with levels 16, 27 and
48, respectively. For X 4 2 and X 6A
2 the L-functions can be written as
LΩ(X d 2 , s) = L(ψ2 d , s), (71)
where ψd are algebraic Hecke characters associated to cusp forms fd(q) of weight two and levels
64 and 27, respectively, given by the elliptic forms
f4(τ) = f(E 4 , q) = η 2 (q 4 )η 2 (q 8 ) ⊗ χ2
f6A(τ) = f(E 3 , q) = η 2 (q 3 )η 2 (q 9 ). (72)
For X 6B
2 the L-series is given by LΩ(X 6B
2 , s) = L(ψ 2 144 ⊗ χ3, s), leading to the cusp form of
level 144
f6B(τ) = f(E 6 , q) = η 4 (q 6 ) ⊗ χ3. (73)
These results will enter in the discussion below of higher dimensional varieties.
7 A nonextremal K3 surface X 12
2 ⊂ P (2,3,3,4)
String theoretic modularity of the class of extremal K3 surfaces of Brieskorn-Pham type has
been established in [11]. Extremal K3 surfaces are characterized by the fact that their Picard
number is maximal, i.e. ρ = 20.
A nonextremal example of a K3 surface is defined as the Brieskorn-Pham hypersurface (1) of
degree twelve in the weighted projective space P(2,3,3,4). The Galois group of the cyclotomic
field Q(µ12) has order four, hence the Ω−motive has rank four. The four Jacobi sums which
parametrize this motive are given by
jp(σαΩ), σ ∈ Gal(Q(µ12)/Q) (74)
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