Buchstaber Genus, Formal Groups, and Cohomology Theories

We discuss the relation between one-dimensional $n$-valued formal groups and the Buchstaber genus. Motivated by Abel’s theorem, we show that $n$-valued formal groups over reduced $\mathbb{Q}$-algebras are automatically commutative and, in the generically separable case, are strictly isomorphic to the elementary cyclic law. Slides Read more

We then construct a one-valued formal group whose modulus square is Buchstaber’s universal two-valued formal group and give explicit formulas for its logarithm and exponential in terms of Weierstrass functions. The associated genus $\mathrm{Bc}$ is compared with the Krichever, Ochanine, and Witten genera.

We also discuss integral values of $\mathrm{Bc}$ on theta divisors and on the Stong ring, a sharper Hurwitz-integrality result for the elliptic sigma-function, and the first obstruction to integrality on spin manifolds in dimension $24$. Finally, we indicate how the Buchstaber genus is realized by complex-oriented cohomology theories.