  We extend the dictionary between the BPS spectrum of Heterotic strings and
the one of F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to
cases with higher rank non-Abelian gauge groups and in particular to dual pairs
between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds
with a compatible genus one fibration. We show how to obtain the new
supersymmetric index purely from the Calabi-Yau geometry by reconstructing the
Noether-Lefschetz generators, which are vector-valued modular forms. There is
an isomorphism between the latter objects and vector-valued lattice Jacobi
forms, which relates them to the elliptic genera and twisted-twined elliptic
genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi
forms generate the dimensions of the refined cohomology of the Hilbert schemes
of symmetric products of the fiber and allow us to refine the BPS indices in
the fiber and therefore to obtain, conjecturally, actual state counts. Using
the properties of the vector-valued lattice Jacobi forms we also provide a
mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory
compactified on this general class of fibered Calabi-Yau 3-folds.
