author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{An Algorithm to Compute Algebraic Relations Between Modular Functions}},
language = {english},
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 $\lambda$-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-G\"ollnitz-Gordon type as another promising area of MultiSamba application. },
number = {25-09},
year = {2025},
month = {November},
keywords = {modular functions, multisamba, modular equations},
length = {23},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}