# Galois Action on the Homology of Fermat Curves

Conference Paper

In his paper titled â€œTorsion points on Fermat Jacobians, roots of circular units and relative singular homology,â€ 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 Vandiverâ€™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 â€“ 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 Vandiverâ€™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