Krichever Formal Groups

V. M. Buchstaber ; E. Yu. Bun'kova

Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 2, pp. 23-44.

Résumé

On the basis of the general Weierstrass model of the cubic curve with parameters $\mu=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_6)$, 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 $\mathbb{Z}[\mu]$ are obtained. The notion of the universal Krichever formal group over the ring $\mathcal{A}_{\mathrm{Kr}}$ is introduced; its exponential is determined by the Baker–Akhiezer function $\Phi(t)=\Phi(t;\tau,g_2,g_3)$, where $\tau$ 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 $\mathcal{A}_{\mathrm{Kr}}\otimes \mathbb{Q}$ 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 $\Psi(t)=\Psi(t; v,w, \mu)$ 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 $\mu$. For $w\neq\pm v$, this function has the branching points $t=v$ and $t=-v$, and for $w=\pm v$, it coincides with $\Phi(t;v,g_2,g_3)$ and becomes meromorphic. An addition theorem for the function $\Psi(t)$ is obtained. According to this theorem, the function $\Psi(t)$ is the common eigenfunction of differential operators of orders 2 and 3 with doubly periodic coefficients.

Détail

Mots-clés : elliptic Hirzebruch genera
Keywords: addition theorems, Baker–Akhiezer function, deformed Lamé equation.

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