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Open Access Publications from the University of California

Stabilization of a Tower of Universal Deformation Rings

  • Author(s): Gim, Geunho
  • Advisor(s): Hida, Haruzo
  • et al.

Given a $ p $-adic absolutely irreducible residual representation of a Galois group with Mazur's finiteness condition $ \Phi_p $, we get a universal deformation ring in Mazur's sense. Consider a tower of intermediate field extensions from the base field and restrictions of the given representation to each intermediate field. If all the restrictions are absolutely irreducible, we get a universal deformation ring associated to each restriction. In this way, we get a tower of universal deformation rings and the morphisms between them that are provided by universality. A natural question to ask here is that whether the tower of universal deformation rings stabilize or not, that is, whether the size of the universal deformation rings stops growing or grows indefinitely over the tower. In this thesis, we answer this question in the case of cyclotomic extensions of a number field.

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