Eisenstein series identities based on partial fraction decomposition

Abstract

From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}{E}_{k-2i}(2\le i\le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\mathrm{SL}}_2(\mathbb{Z})$.

arXiv comment8 pages