Genera defined by hyperelliptic integrals and Siegel modular functions

We study genera defined by hyperelliptic integrals. The associated formal group laws select a particular set of rational generators of the complex cobordism ring. Hyperelliptic genera and their kernels are concisely described in terms of these generators. Using Thomaeʹs formula which expresses branch points of hyperelliptic curves in terms of hyperelliptic theta constants, we express values of hyperelliptic genera in terms of hyperelliptic theta constants evaluated at period matrices of the associated hyperelliptic curves. In particular, in genus 2 case, we obtain a hyperelliptic genus whose values lie in the ring of level 2 genus 2 Siegel modular functions.