Explicit coverings of families of elliptic surfaces by squares of curves

On Kummer‐like surfaces attached to singularity and modular forms

“…Also, the Hermitian modular forms 𝑡 𝑗 (𝑗 ∈ {4, 6, 10, 12, 18}) have exact expression using theta functions on the symmetric domain for 𝑆𝑈(2, 2) and invariants of a complex reflection group (see [15]). Furthermore, it seems to the authors that the results of the recent work [8], in which motives of 𝐾3 surfaces are studied, are closely related to our surfaces 𝐾(𝑡) and 𝑆(𝑡). Hence, the authors expect that those properties of our families provide new merits for future research of arithmetic theory of 𝐾3 surfaces.…”

Hyper–Kähler Manifolds of Generalized Kummer Type and the Kuga–Satake Correspondence

“…In [ 5 ], the Kuga–Satake Hodge conjecture is proved for K3 surfaces which are desingularization of singular K3 surfaces in with 15 nodal points. The authors then show that the same techniques as in Theorem 2.16 can be used to prove the Hodge conjecture for the square of these K3 surfaces.…”

“…[ 5 ] Let X be a K3 surface which is the desingularization of a singular K3 surface in with 15 nodal points. Then, the Kuga–Satake Hodge conjecture holds for X and the Hodge conjecture is true for .…”

“…The families of K3 surfaces studied in [ 14 ] and in [ 5 ] satisfy the hypotheses of Theorem 2.18 . Therefore, Theorem 2.18 shows in particular that the Hodge conjecture holds for all powers of the K3 surfaces in the two families considered in [ 5 , 14 ].…”

“…The families of K3 surfaces studied in [ 14 ] and in [ 5 ] satisfy the hypotheses of Theorem 2.18 . Therefore, Theorem 2.18 shows in particular that the Hodge conjecture holds for all powers of the K3 surfaces in the two families considered in [ 5 , 14 ]. The techniques applied are similar to the one introduced in [ 14 ] with the addition of a deformation argument which allows to prove the Hodge conjecture for all powers of the K3 surfaces of higher Picard number in the family.…”