A family of Étale coverings of the affine line

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Abstract

It this note we prove the following theorem. Let Πalg1(A1C) be the algebraic fundamental group of the affine line over C, where C is the completion of the algebraic closure of Fq((1/T)), and Fq is a field with q elements. If Fq has at least four elements, then we show that there is a continuous surjection Πalg1(A1C) → lim SL2(A/I)/{+ ± 1}, where A = Fq[T] and the inverse limit is over the family of non-zero, proper ideals of A. This result is proved by using the moduli of Drinfel'd A-modules of rank two over C with I-level structures; these curves give (tamely) ramified covers of the line and the tame ramification is removed using a variant of Abhyankar's lemma.

Original languageEnglish (US)
Pages (from-to)414-418
Number of pages5
JournalJournal of Number Theory
Volume59
Issue number2
DOIs
StatePublished - Aug 1996
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory

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