Elementary proofs of relations between Eisenstein series

Abstract

SynopsisEisenstein series are entire modular forms Ek of even integral weight k≧ 4 with Fourier expansions given by (1.1). There are numerous identities, such as E8 = , relating these series. These are usually proved by arguments making use of the dimensions of vector spaces of modular forms, and not directly. The paper shows how such identities can be proved by elementary methods by studying chains of solutions of Diophantine equations of the form xξ+yη = n.