Ramanujan-Shen's differential equations for Eisenstein series of level 2

TL;DR

This paper derives new differential equations for Eisenstein series of level 2 using Jacobi theta functions, providing a novel characterization and applications to Ramanujan's tau function.

Contribution

It introduces new differential equations for Eisenstein series of level 2, expanding the understanding of their properties and connections to Ramanujan's tau function.

Findings

New differential equations for Eisenstein series of level 2

Characterization of these equations via Jacobi theta functions

Arithmetic results on Ramanujan's tau function

Abstract

Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. This gives a new characterization of a system of differential equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike (2003), Maier (2011) and Toh (2011). As application, we show some arithmetic results on Ramanujan's tau function.