Flat structure for the simple elliptic singularity of type $\widetilde {\bf E_6}$ and Jacobi form

TL;DR

This paper constructs explicit flat theta invariants for the simple elliptic singularity of type E6 using Jacobi forms, aiding in solving the period mapping inversion problem for this singularity.

Contribution

It explicitly constructs flat theta invariants for type E6 elliptic singularities using Jacobi forms, advancing the understanding of period mappings and flat structures.

Findings

Explicit flat theta invariants for type E6

Solution to Jacobi's inversion problem for this singularity

Connection between flat structures and Jacobi forms

Abstract

In order to construct the inverse mapping of the period mapping for the primitive form for the semi-universal deformation of a simple elliptic singularity, K.Saito introduced in [5] the ``flat structure'' for the extended affine root system. In section 3, we construct explicitly the flat theta invariants in the case of type $E_6$ using the Jacobi form introduced by Wirthm\"uller [7]. Combining the results of Kato [3], Noumi [4] (explicit description of the flat coordinates), this gives an answer to Jacobi's inversion problem (up to linear isomorphism) of this period mapping for a simple elliptic singularity of type $\tilde E_6$ (see also [6]). The details will be published elsewhere.