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A Family of Étale Coverings of the Affine Line

✍ Scribed by Kirti Joshi

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
223 KB
Volume
59
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

It this note we prove the following theorem. Let ? alg 1 (A 1 C ) be the algebraic fundamental group of the affine line over C, where C is the completion of the algebraic closure of F q ((1ÂT)), and F q is a field with q elements. If F q has at least four elements, then we show that there is a continuous surjection ? alg 1 (A 1 C ) Ä SL 2 (AÂI)Â[+ \1], where A=F q [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.

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