## Galois Action on the Homology of Fermat Curves

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.

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*Galois Action on the Homology of Fermat Curves*. Vol. 3, 2016, pp. 57–86.

*Scopus*, doi:10.1007/978-3-319-30976-7_3.