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