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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Deformation theory and the tame fundamental group
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by David Harbater PDF
Trans. Amer. Math. Soc. 262 (1980), 399-415 Request permission


Let U be a curve of genus g with $n + 1$ points deleted, defined over an algebraically closed field of characteristic $p \geqslant 0$. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite $p’$-groups on $n + 2g$ generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin’s Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 399-415
  • MSC: Primary 14E20; Secondary 14D15, 14H30
  • DOI:
  • MathSciNet review: 586724