Krichever formal groups

Abstract

On the basis of the general Weierstrass model of the cubic curve with parameters µ = (µ 1 , µ 2 , µ 3 , µ 4 , µ 6 ), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in Z[µ] are obtained.The notion of the universal Krichever formal group over the ring A Kr is introduced; its exponential is determined by the Baker-Akhiezer function Φ(t) = Φ(t; τ, g 2 , g 3 ), where τ is a point on the elliptic curve with Weierstrass parameters (g 2 , g 3 ). As a consequence, results on the Krichever genus which takes values in the ring A Kr ⊗ Q of polynomials in four variables are obtained. Conditions necessary and sufficient for an elliptic formal group to be a Krichever formal group are found.A quasiperiodic function Ψ(t) = Ψ(t; v, w, µ) is introduced; its logarithmic derivative defines the exponential of the general elliptic formal group law, where v and w are points on the elliptic curve with parameters µ. For w = ±v, this function has the branching points t = v and t = −v, and for w = ±v, it coincides with Φ(t; v, g 2 , g 3 ) and becomes meromorphic. An addition theorem for the function Ψ(t) is obtained. According to this theorem, the function Ψ(t) is the common eigenfunction of differential operators of orders 2 and 3 with doubly periodic coefficients.

On the Buchstaber formal group law and some related genera

“…Note that Proposition 3.1 agrees with the results in [8] on the structure of the coefficient ring of F Kr obtained in terms of the associativity equation. The new information here concerning the Krichever group, and hence the Krichever genus, is that in dimensions 20 and 24 there are no indecomposable elements, because z 10 = 0 and z 12 = 0.…”

“…It is also proved in [8] that b(x) = ∂F Kr ∂y (x, 0). Following [6], in [10] the authors considered the following formal group law corresponding to the Krichever genus:…”

“…In [14,16] the general four-variable complex elliptic genus φ KH on MU * ⊗ Q is defined to be determined by the following property: if one denotes by f Kr (x) the exponential of the Krichever universal formal group law F Kr [8], then the series…”

“…The formal group F Kr is a particular case of F KH corresponding to the genus φ KH , as (7.1) holds for F Kr (see [8]). Therefore, rationally these formal groups coincide and have the coefficient ring…”

“…Clearly, 8 to calculate the values of the ψ-genus on CP i , the generators of the rational complex bordism ring MU * ⊗ Q = Q[CP 1 , CP 2 , . .…”