Fake proofs for identities involving products of Eisenstein series

TL;DR

This paper presents informal, 'fake' proofs of identities involving products of Eisenstein series, highlighting intriguing patterns despite their lack of mathematical rigor and the need to understand the underlying phenomena.

Contribution

It introduces a novel, informal proof approach for identities of Eisenstein series, stimulating further investigation into their underlying structure and convergence issues.

Findings

Fake proofs suggest interesting patterns in Eisenstein series identities

Some fake proofs produce suggestive but incorrect results

Highlights the need to understand convergence and validity in such identities

Abstract

In the workshop of the July 2016 Building Bridges 3 conference in Sarajevo, I presented the results from a joint article with W. Raji (Mathematische Annalen 2017, preprint arXiv:1402.1854). That article gave a proof of various linear relations between products of two Eisenstein series on $\Gamma(N)$, including an interesting identity related to the action of a Hecke operator on such a product. The real proofs involve some care to deal with issues of convergence. In this note we give "fake" proofs for these identities, ignoring the convergence issues; some of these fake proofs appeared in the workshop lecture as an amusing side note before I sketched the real proofs. Something in these fake proofs is quite suggestive, even though the proofs themselves are clearly invalid (and even produce wrong results). It would be interesting to understand what exactly is going on here.