Galois action on the homology of Fermat curves

الأصل · EN

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 GQ₍ζₙ₎. In particular, when n is an odd prime p, he shows that the action of GQ₍ζₚ₎ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial 1-(1-xᵖ)ᵖ. If p satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over Q(ζₚ) is an elementary abelian p-group of rank (p+1)/2. Using an explicit basis for this Galois group, 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.