  In this paper, we consider natural geometric objects coming from Lagrangian
Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit
Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially
classical modular forms. Recent work by Zwegers and two of the authors
determined modularity properties of several simpler pieces of the last, and
most mysterious, function by developing several identities between functions
with properties generalizing those of the mock modular forms in Zwegers'
thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and
Lau's functions, inspired by recent work of Alexandrov, Banerjee, Manschot, and
Pioline on similar functions. Combined with the work of Lau and Zhou, as well
as the aforementioned work of Zwegers and two of the authors, this affords a
complete understanding of the modularity transformation properties of the open
Gromov-Witten potentials of elliptic orbifolds of the form $\mathbb
P^1_{a,b,c}$ computed by Cho, Hong, Kim, and Lau. It is hoped that this will
provide a fuller picture of the mirror-symmetric properties of these orbifolds
in subsequent works.
