  A fundamental object of study in mirror symmetry of $n$-dimensional Fano
varieties is the A-side connection on small quantum cohomology. When the Picard
rank is 1, the Borel transform relates the quantum differential operator of the
Fano to the Picard-Fuchs operator of the mirror to the associated pencil of
anticanonical Calabi-Yau $(n-1)$-folds on the Fano variety. Expanding on
related work by W. Yang on the Beukers-Peters pencil of K3 surfaces associated
with Ap\'ery's proof for the irrationality of $\zeta(3)$, for such operators we
define holomorphic prepotentials, virtual Yukawa couplings, and virtual
instanton numbers, analogous to the usual ingredients of Calabi-Yau mirror
symmetry. We prove that when the underlying Calabi-Yau operator is modular, the
virtual Yukawa coupling is a modular form of weight-$(n+1)$, with the
holomorphic prepotential as an Eichler integral. We then analyze the quantum
differential operators for modular pencils of K3 surfaces arising as
Dolgachev-Nikulin-Pinkham mirrors for the anticanonical linear systems for the
17 deformation classes of Fano threefolds of Picard rank-1 classified by
Iskovskikh from the perspective of Golyshev & Zagier's proof of the Gamma
conjecture for such Fanos, the natural setting of Yang's work. Here, the
virtual instanton numbers are proven to be periodic integers with period equal
to the level of the modular subgroup. Finally, we conjecture that the geometric
nature of these virtual instanton numbers can be understood in terms of
relative genus zero Gromov-Witten invariants of the associated log Calabi-Yau
pair from the Fano threefold and anticanonical K3.
