Universal Formal Group for Elliptic Genus of Level N

Abstract

An elliptic function of level N determines an elliptic genus of level N as a Hirzebruch genus. It is known that any elliptic function of level N is a specialization of the Krichever function that determines the Krichever genus. The Krichever function is the exponential of the universal Buchstaber formal group. In this work we give a specialization of the Buchstaber formal group such that this specialization determines formal groups corresponding to elliptic genera of level N. Namely, an elliptic function of level N is the exponential of a formal group of the form F(u, v) = (u2A(v) − v2A(u))/(uB(v) − vB(u)), where A(u), B(u) ∈ ℂ[[u]] are power series with complex coefficients such that A(0) = B(0) = 1, A″(0) = B′(0) = 0, and for m = [(N − 2)/2] and n = [(N − 1)/2] there exist parameters (a1, …, am, b1, …, bn) for which the relation \(\prod\nolimits_{j=1}^{n-1}\left(B(u)+b_{j} u\right)^{2} \cdot\left(B(u)+b_{n} u\right)^{N-2 n}=A(u)^{2} \prod\nolimits_{k=1}^{m-1}\left(A(u)+a_{k} u^{2}\right)^{2} \cdot\left(A(u)+a_{m} u^{2}\right)^{N-1-2 m}\) holds. For the universal formal group of this form, the exponential is an elliptic function of level at most N. This statement is a generalization to the case N > 2 of the well-known result that the elliptic function of level 2 determining the elliptic Ochanine–Witten genus is the exponential of a universal formal group of the form F(u, v) = (u2 − v2)/(uB(v) − vB(u)), where B(u) ∈ ℂ[[u]], B(0) = 1, and B′(0) = 0. We prove this statement for N = 3, 4, 5, 6. We also prove that the elliptic function of level 7 is the exponential of a formal group of this form. Universal formal groups that correspond to elliptic genera of levels N = 5, 6, 7 are obtained in this work for the first time.