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On nearly ordinary Hecke algebras for $$GL(2)$$ over totally real fields. (English) Zbl 0742.11026
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 139-169 (1989).
[For the entire collection see Zbl 0721.00006.]
Let $$F$$ be a totally real field of degree $$d$$, $$p$$ a rational prime, and $${\mathcal O}$$ a valuation ring finite and flat over $$\mathbb{Z}_ p$$ and containing all conjugates of the ring of integers of $$F$$. In earlier work the author has constructed, for each positive weight $$v$$, a Hecke algebra $$h_ v^{ord}(1,{\mathcal O})$$ such that for each non-negative weight $$n$$ parallel to $$-2v$$ the Hecke algebra over $${\mathcal O}$$ for the space of Hilbert cusp forms of level $$p^ \alpha$$ and weight $$(n+2t,v+n+t)$$ can be obtained uniquely as a quotient of $$h_ v^{ord}(1,{\mathcal O})$$. In this paper the author constructs a universal Hecke algebra of which each algebra $$h_ v^{ord}(1,{\mathcal O})$$ is a quotient.
As a corollary, the author observes that if $$f$$ is a normalized Hilbert eigenform then $$f$$ has an $$s$$-dimensional $$p$$-adic deformation over $${\mathcal O}$$, where $$s>d$$. Moreover, if Leopoldt’s conjecture holds for $$F$$ and $$p$$ then, in fact, $$s=d+1$$.

MSC:
 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11S40 Zeta functions and $$L$$-functions