In a previous post, I began discussing a theorem of Ochanine:

Theorem 1 (Ochanine)A genus annihilates the projectivization of every even-dimensional complex bundle if and only if the logarithm of is an elliptic integral

In the previous post, we described Ochanine’s proof that a genus whose logarithm is an elliptic integral (a so-called *elliptic genus*) annihilated any such projectivization. The proof relied on some computations in the projectivization and then some trickery with elliptic functions. The purpose of this post is to prove the converse: a genus with a suitably large kernel comes from an elliptic integral. (more…)