Galois Action on the Homology of Fermat Curves

Abstract

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 \(G_{\mathbb{Q}(\zeta _{n})}\). In particular, when n is an odd prime p, he shows that the action of \(G_{\mathbb{Q}(\zeta _{p})}\) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial \(1 - (1 - x^{p})^{p}\). If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over \(\mathbb{Q}(\zeta _{p})\) is an elementary abelian p-group of rank \((p + 1)/2\). 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 \(G_{\mathbb{Q}(\zeta _{n})}\). In particular, when n is an odd prime p, he shows that the action of \(G_{\mathbb{Q}(\zeta _{p})}\) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial \(1 - (1 - x^{p})^{p}\). If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over \(\mathbb{Q}(\zeta _{p})\) is an elementary abelian p-group of rank \((p + 1)/2\). 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.