Zeta functions of an infinite family of K 3 surfaces

Abstract

We identify a parameterized family of K 3 surfaces with generic Picard number 19, and we employ elementary methods to determine their local zeta functions. In addition, we explicitly determine those surfaces which are modular.

ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

“…The arithmetic relation between X λ and E λ is obtained by Ahlgren, Ono, and Penniston [17]. The j−invariant of E λ is…”

On Shioda–Inose structures of one-parameter families of K3 surfaces

“…Let E (1) and E (2) be two elliptic curves defined over a certain field K which may be a number field, a finite field, or a function field, etc. An isogeny between E (1) and E (2) over K is a non-trivial over K morphism : E (1) −→E (2) .…”

“…Suppose E (1) and E (2) are two elliptic curves given by equations of the form of (13) over C(t) with non-constant j -functions. If there is an isogenous map : E (1) − →E (2) , then the Picard-Fuchs equations satisfied by E (1) and E (2) respectively are equivalent. (14) be holomorphic 1-forms on E (1) and E (2) , respectively.…”

“…It can be also used to prove some identities involving character sums. In a paper of Ahlgren et al [1], by using pure character sum calculations the authors prove a certain family of elliptic curves E t is related to a family of K3 surfaces X t by a S-I structure. But they do not reveal how these families are discovered.…”

“…But they do not reveal how these families are discovered. In a recent paper [11], the author utilizes this algorithm to explain how to construct explicitly the j -function of E t once the family X t in [1] is given.…”