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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, \(s=d+1\).

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