Equations D3 and spectral elliptic curves

TL;DR

This paper investigates modular differential equations of orders 2 and 3, revealing their solutions' connection to spectral elliptic curves and identifying equations with multiplicativity properties.

Contribution

It introduces a classification of D3 equations with multiplicativity and links them to spectral elliptic curves and Zagier's D2 equations.

Findings

Complete list of D3 equations with multiplicativity property

Solutions match q-expansions of spectral elliptic curves

Relation established between D3 and D2 equations

Abstract

We study modular determinantal differential equations of orders 2 and 3. We show that the expansion of the analytic solution of a non-degenerate modular equation of type D3 over the rational numbers with respect to the natural parameter coincides, under certain assumptions, with the q-expansion of the newform of its spectral elliptic curve and therefore possesses a multiplicativity property. We compute the complete list of D3 equations with this multiplicativity property and relate it to Zagier's list of non-degenerate modular D2 equations.