  We study the enumerative geometry of algebraic curves on abelian surfaces and
threefolds. In the abelian surface case, the theory is parallel to the
well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We
prove complete results in all genera for primitive classes. The generating
series are quasimodular forms of pure weight. Conjectures for imprimitive
classes are presented. In genus 2, the counts in all classes are proven.
Special counts match the Euler characteristic calculations of the moduli spaces
of stable pairs on abelian surfaces by G\"ottsche-Shende. A formula for
hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a
transversality statement).
  For abelian threefolds, complete conjectures in terms of Jacobi forms for the
generating series of curve counts in primitive classes are presented. The base
cases make connections to classical lattice counts of Debarre, Goettsche, and
Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition
function computations for abelian threefolds. A multiple cover structure is
presented. The abelian threefold conjectures open a new direction in the
subject.
