Universal Deformations, Rigidity,and Ihara's Cocycle

Abstract

Let 𝒜 be the ring ℤ p [[t 0, t 1, t ∞]]/((t 0 + 1)(t 1 + 1)(t ∞ + 1) − 1)equipped with the non-trivial action of G ℚ ≔ Gal(ℚ¯/ℚ) described in the introduction. In Ihara (1986b), Ihara constructs a universal cocycle arising from the action of Gal(ℚ¯/ℚ) on certain quotients of the Jacobians of the Fermat curves for each n ≥ 1. This paper gives a different construction of part of Ihara's cocycle by considering the universal deformation of certain two-dimensional representations of Πℚ¯, the algebraic fundamental group of ℙ1(ℚ¯)\\{0, 1, ∞}. More precisely, we determine the universal deformation ring subject to certain deformation conditions arising from a residual representation Belyĭ's Rigidity Theorem is then used to extend each such universal deformation to a representation of Π K , where K is a finite cyclotomic extension of ℚ(μ p ∞ ). When ρ¯ is the representation arising from the p-division points of the Legendre family of elliptic curves, we give a geometric construction of one such extended universal deformation ρ, and show that part of Ihara's cocycle can be recovered by specializing ρ at infinity.