See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here.
Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative property.
Property. For any appropriate fiber bundle $F \to E \to B$ of manifolds, we have$$\phi(E) = \phi(B)\pi(F).\tag*{$(1)$}$$When $B$ is simply connected, this is true for the signature by an old theorem of Chern, Hirzebruch, and Serre.
A special case of the property $(1)$ is that whenever $E \to B$ is an even-dimensional complex vector bundle, then we have$$\phi(\mathbb{P}(E)) = 0,$$for $\mathbb{P}(E)$ the projectivization: this is because $\mathbb{P}(E) \to B$ is a fiber bundle whose fibers are odd-dimensional complex projective spaces, which vanish in the cobordism ring.
Ochanine has given a complete characterization of the genera which satisfy this property.
Theorem 1 (Ochanine). A genus $\phi$ annihilates the projectivizations $\mathbb{P}(E)$ of even-dimensional complex vector bundles if and only if the associated log series$$g(x) = \sum {{\phi\left(\mathbb{CP}^{2i}\right)}\over{2i + 1}}x^{2i+1}$$is given by an elliptic integral$$g(x) = \int_0^x Q(u)^{-1/2}du,$$for $Q(u) = 1 - 2\delta u^2 + \epsilon u^4$ for $\delta$, $\epsilon$.
Akhil gives a proof, and at the end, he remarks, and I ask the following question(s).
I must confess that this proof still feels like magic to me; it's not at all clear what is really going on, and I can't really tell what to take away from it. Does anyone reading this have any ideas? Is there a "high-concept" explanation for this result?
Thanks in advance.
