Abstract
This paper proves the remaining open case of Abhyankar’s higher dimensional conjecture on local fundamental groups in characteristic p ([Ab2], [Ab3]). This conjecture, which is analogous to Abhyankar’s conjectures on global fundamental groups, proposed that a finite group G is a Galois group over k[[x 1,…, x n]][(x 1… x r)-1] if and only if its maximal prime-to-p quotient is, provided n≥ 2 and 1 ≤ r ≤ n. For r > 1, this conjecture was disproven in [HP]. Here we prove that the conjecture is true in the case r = 1. So the Galois groups over k[[x 1,…, x n]][x 1 -11] are precisely the cyclic-by-quasi-p groups.
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Harbater, D., Stevenson, K.F. (2004). Abhyankar’s Local Conjecture on Fundamental Groups. In: Christensen, C., Sathaye, A., Sundaram, G., Bajaj, C. (eds) Algebra, Arithmetic and Geometry with Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18487-1_26
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DOI: https://doi.org/10.1007/978-3-642-18487-1_26
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