Krichever formal groups

V. M. Buchstaber, E. Yu Bunkova

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

Abstract

On the basis of the general Weierstrass model of the cubic curve with parameters μ = (μ1, μ2, μ3, μ4, μ6), 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 AKr is introduced; its exponential is determined by the Baker-Akhiezer function Φ(t) = Φ(t; τ, g2, g3), where τ is a point on the elliptic curve with Weierstrass parameters (g2, g3). As a consequence, results on the Krichever genus which takes values in the ring AKr ⊗ 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, g2, g3) 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.

Original languageEnglish
Pages (from-to)99-116
Number of pages18
JournalFunctional Analysis and its Applications
Volume45
Issue number2
DOIs
Publication statusPublished - 8 Mar 2011
Externally publishedYes

Keywords

  • addition theorems
  • Baker-Akhiezer function
  • deformed Lamé equation
  • elliptic Hirzebruch genera

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