Elliptic formal group laws, integral Hirzebruch genera and Krichever genera

TL;DR

This paper explores the structure of elliptic formal group laws, their relation to Hirzebruch and Krichever genera, and introduces a new genus based on a generalized Baker-Akhiezer function with unique properties.

Contribution

It analyzes the elliptic formal group law over polynomial rings and introduces a new Krichever genus using a generalized Baker-Akhiezer function with novel features.

Findings

Describes the structure of the elliptic formal group law.

Introduces a 5-parametric family of Hirzebruch genera with integer values.

Defines a new Krichever genus based on a generalized Baker-Akhiezer function.

Abstract

We consider the geometrical addition law on the elliptic curve in Tate coordinates. It corresponds to the general formal group law over the ring of polynomials with integer coefficients of the parametra of the curve. We study the structure of this law and the differential equation that determines its exponent. We describe a 5-parametric family of Hirzebruch genera with integer values on stably complex manifolds. We introduce the general Krichever genus, which is given by a generalized Baker-Akhiezer function. This function has many of the fundamental properties of the Baker-Akhiezer function, but unlike it, it is not meromorphic, because it can have two branch points in the parallelogram of periods.