Galois Action on the Homology of Fermat Curves

Published

Conference Paper

© Springer International Publishing Switzerland 2016. In his paper titled “Torsion points on Fermat Jacobians, roots of circular units and relative singular homology,” Anderson determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group (Forumala presented). In particular, when n is an odd prime p, he shows that the action of (Forumala presented). on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial (Forumala presented). If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over (Forumala presented). is an elementary abelian p-group of rank (Forumala presented). Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points. In Anderson (Duke Math J 54(2):501 – 561, 1987), the author determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group (Forumala presented). In particular, when n is an odd prime p, he shows that the action of (Forumala presented). on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial (Forumala presented). If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over (Forumala presented). is an elementary abelian p-group of rank (Forumala presented). Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.

Full Text

Duke Authors

Cited Authors

  • Davis, R; Pries, R; Stojanoska, V; Wickelgren, K

Published Date

  • January 1, 2016

Volume / Issue

  • 3 /

Start / End Page

  • 57 - 86

Electronic International Standard Serial Number (EISSN)

  • 2364-5741

International Standard Serial Number (ISSN)

  • 2364-5733

Digital Object Identifier (DOI)

  • 10.1007/978-3-319-30976-7_3

Citation Source

  • Scopus