The Galois action and cohomology of a relative homology group of Fermat curves

For an odd prime p satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group GQ(ζp) on the homology of the degree p Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.