The Galois action and cohomology of a relative homology group of Fermat curves (Q1644991)

Let \(p\) be a prime satisfying Vandiver's conjecture, i.e., such that \(p\) does not divide the order of \(h^+\) of the class group of \(\mathbb{Q}(\zeta+\zeta^{-1})\), where \(\zeta\) is a \(p\)-th root of unity. Let \(X\) be the degree \(p\) Fermat curve \(x^p+y^p=z^p\). Let \(U\subset X\) be the affine open given by \(z\neq 0\). Consider the closed subscheme \(Y\subset U\) defined by \(xy=0\). Let \(H_1(U,Y;\mathbb{Z}/p)\) denote the étale homology group with \(\mathbb{Z}/p \) coefficients, of the pair \((U\otimes \bar{K},Y\otimes\bar{K})\). By [\textit{G. W. Anderson}, Duke Math. J. 54, 501--561 (1987; Zbl 1370.11069)], the group \(H_1(U,Y;\mathbb{Z}/p)\) is a free rank-one \(\mathbb{Z}/p[\mu_p\times\mu_p]\)-module with generator \(\beta\). The Galois action of \(\sigma\in G_{\mathbb{Q}(\zeta)}\) is then determined by \(\sigma\beta=B_\sigma\beta\), for some \(B_\sigma\in \mathbb{Z}/p[\mu_p\times\mu_p]\). Anderson theoretically described \(B_\sigma\). In this paper, a closed form formula for \(B_\sigma\) is given. Intermediate results by the same authors [\textit{R. Davis} et al., Assoc. Women Math. Ser. 3, 57--86 (2016; Zbl 1416.11045)] about the isomorphism class of the Galois group of the field extension through the action of \(G_{\mathbb{Q}(\zeta)}\) factors, are strongly used. The first application of this formula is that the norm of \(B_\sigma\) is \(0\) for almost all \(\sigma\). This is important in computing Galois cohomology as in Section 6 where a method for the efficient computation of the first cohomology group \(H^1(G_{\mathbb{Q}(\eta)}, H_1(U,Y;\mathbb{Z}/p))\) is given. This will eventually play a key role in understanding obstructions for rational points on Fermat curves as Ellenberg's obstruction related to the non-abelian Chabauty method. A second application of the main formula is a proof of the fact that \(H_1(U;\mathbb{Z}/p)\) is trivialized by the product of \(\lfloor 2p/3\rfloor\) terms of the form \((B_\sigma-1)\).