Spectral curve underlying the Apéry sequence

Identify the hidden spectral curve determined by the Apéry sequence and its associated Picard–Fuchs equation, with the aim of uncovering the geometric structure behind the Apéry numbers and the irrationality of ζ(3).

Background

The paper studies the Apéry sequence through its generating function, a Picard–Fuchs equation associated with a family of K3 surfaces, and its relation to genus-zero, one-point Gromov–Witten invariants of the Fano threefold V12. In the abstract, the author identifies the discovery of a hidden curve behind the Apéry sequence as an unresolved goal. The later discussion suggests that this curve should be determined by the type (0,1) Gromov–Witten invariants of V12 and should correspond, on the B-model side, to the relevant Picard–Fuchs equation.