We parameterize the Neron-Severi group of certain abelian surfaces A in terms of binary quadratic forms. The parameterization leads to simple criteria for A to contain a smooth curve of any fixed genus and for A to admit a very ample line bundle of any fixed degree. We show that there are only finitely many (moduli spaces of) such surfaces which do not embed in P^4. Assuming the generalized Riemann hypothesis, one can specify exactly which A have this property. This is joint work with Julian Rosen.