Abstract
Abstract This paper concerns the study of the Schwartz differential equation { h , τ } = s E 4 ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s E 4 y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .
Highlights
Let D be a domain in C and f a meromorphic function on D
This paper concerns the study of the Schwartz differential equation {h, τ} = s E4(τ), where E4 is the weight 4 Eisenstein series and s is a complex parameter
This leads to the solutions to the Fuchsian differential equation y + s E4 y = 0
Summary
Let D be a domain in C and f a meromorphic function on D. This allows us to explicitly write down solutions in terms of classical modular forms and functions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
