Belyi map verification using certified path tracking

TL;DR

This paper introduces a rigorous workflow for computing and certifying the monodromy of Belyi maps using certified homotopy continuation, applied at scale to the LMFDB.

Contribution

It presents a novel end-to-end method for verifying Belyi maps' monodromy over number fields with certified path tracking, enabling large-scale certification.

Findings

Successfully certified monodromy triples for numerous Belyi maps in LMFDB

Demonstrated the effectiveness of certified homotopy continuation in algebraic geometry

Provided a scalable approach for verifying complex algebraic structures

Abstract

We provide an end-to-end workflow to rigorously compute the monodromy of Belyi maps from exact equations over number fields using certified homotopy continuation. We then apply this method at scale to certify the monodromy triples of Belyi maps in the $L$-functions and Modular Forms Database (LMFDB).