Galois module structure and Jacobians of Fermat curves

Abstract

The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.

Galois Action on the Homology of Fermat Curves

“…Information on fields generated by points of the Jacobian of quotients of Fermat curves is also contained in [CNGJ13], [CTT98], [Gre81], and [Tze07].…”

“…Thus computations of H 1 (G S , π ab ⊗ Z/p) and H 2 (G S , (π ab ∧ π ab ) ⊗ Z/p) give information about rational points. Groups closely related to H 1 (G S , π ab ⊗ Z/p) also appear in [CNGJ13] and [McC94].…”

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

“…The work of [CNGJ13] gives interesting information related to the embedding Jac X(k) ⊂ H 1 (G, π ab p ) for the Fermat curve. To pursue this application in the case of Fermat curves, set M = H 1 (U, Y ; Z/p) and Q = Gal(L/K).…”