An algorithm to compute algebraic relations between modular functions

Abstract The existence of an algebraic relation between two modular functions, in short: a modular equation, is implied by a classical fact from the theory of compact Riemann surfaces. In this article, we present a new, purely algebraic proof of the existence of modular equations. Our setting consists of an algorithmic framework which is based on a reduction procedure for tuples of formal Laurent series. The resulting algorithm MultiSamba (“sub-algebra module basis algorithm”) is part of Hemmecke’s computer algebra package QEta which has been implemented in FriCAS, a general purpose computer algebra system which is freely available as open source. QEta is a powerful tool-box for actual computations. For example, MultiSamba has been used for computer-assisted discovery and proofs of Ramanujan-Sato series. In this article, we describe the mathematics underlying the MultiSamba algorithm. Moreover, we explain in detail how MultiSamba works for the derivation of a well-known modular equation between the modular $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -function and the Klein j function. Other examples of the automatic discovery and proving of modular equations include identities by Alladi and others, which suggest relations of Ramanujan-Göllnitz-Gordon type as another promising area of MultiSamba application.