BCOV rings on elliptic curves and eta function

TL;DR

This paper explores BCOV rings on elliptic curves, showing they are parametrized by eta function q-exponents, using rational solutions of Riccati equations.

Contribution

It introduces a new parametrization of BCOV rings on elliptic curves via eta function q-exponents and classifies rational solutions of related Riccati equations.

Findings

BCOV rings are parametrized by eta function q-exponents

Classification of rational solutions of Riccati equations

Connection between elliptic curves and eta function parameters

Abstract

Associated Legendre functions of the first kind give a family of BCOV rings on elliptic curves. We prove that the family is parametrized by $q$-exponents of the eta function $\eta(q^{24})$. Our method involves a classification of rational solutions of a Riccati equation under some constraints.