The classifying element for quotients of Fermat curves

Abstract

Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and $\infty$. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of $\mathbb{Q}$.

Keywords:

• Absolute Galois group
• Belyi curve
• classifying element
• covering
• curve
• Fermat curve
• fundamental group
• Galois module
• graded Lie algebra
• homology
• lower central series
• modular symbol

2020 Mathematics Subject Classification:

• Primary: 11D41
• 11G32
• 14F35
• 14H30
• 17B70
• Secondary: 11F06
• 11F67
• 11G30
• 13A50
• 14F20

Acknowledgments

We would like to thank the anonymous referee for their thoughtful and constructive feedback.

Additional information

Funding

Duque-Rosero was supported by a grant from the Simons Foundation (550029, to Voight). Pries was supported by NSF grant DMS-22-00418.