Tamely ramified covers of the projective line with alternating and symmetric monodromy

Renee Bell, Jeremy Booher, William Y. Chen, Yuan Liu

Research output: Contribution to journalArticlepeer-review


Let k be an algebraically closed field of characteristic p and X the projective line over k with three points removed. We investigate which finite groups G can arise as the monodromy group of finite étale covers of X that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime p ≥ 5, there are families of tamely ramified covers with monodromy the symmetric group Sn or alternating group An for infinitely many n. These covers come from the moduli spaces of elliptic curves with PSL2 ( )-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo ℓ.

Original languageEnglish (US)
Pages (from-to)393-446
Number of pages54
JournalAlgebra and Number Theory
Issue number2
StatePublished - 2022
Externally publishedYes


  • characteristic p
  • covers of curves
  • finite fields
  • Markoff triples
  • tame fundamental group
  • tamely ramified covers

ASJC Scopus subject areas

  • Algebra and Number Theory


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