  We compute the Donaldson-Thomas invariants of a local elliptic surface with
section. We introduce a new computational technique which is a mixture of
motivic and toric methods. This allows us to write the partition function for
the invariants in terms of the topological vertex. Utilizing identities for the
topological vertex proved in arXiv:1603.05271, we derive product formulas for
the partition functions. The connected version of the partition function is
written in terms of Jacobi forms. In the special case where the elliptic
surface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula for
primitive curve classes which is independent of the computation of
Kawai-Yoshioka.
