Deformations of induced Galois representations. (English) Zbl 1041.11039

Author’s summary: This article studies the deformation spaces attached to residual representations arising from certain theta-series attached to real quadratic fields. Our interest in the subject stemmed from recent conjectures of H. Hida, K. Doi, and H. Ishii [Conjecture 3.8 of Invent. Math. 134, 547–577 (1998; Zbl 0924.11035)] and H. Hida [Conjecture 2.2 of Doc. Math., J. DMV 3, 273–284 (1998; Zbl 0923.11084)], and others on the behavior of Hecke algebras under cyclic base-change to a totally real field. We are able to verify these conjectures for quadratic base change, under certain hypotheses. The principal techniques used are a study of deformation and pseudo-deformation functors, the latter being inspired by the work of A. J. Wiles and C. M. Skinner [Proc. Natl. Acad. Sci. USA 94, 10520-10527 (1997; Zbl 0924.11044); Publ. Math., Inst. Hautes √Čtud. Sci. 89, 5–126 (1999; Zbl 1005.11030)].


11F80 Galois representations
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