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$$p$$-adic Hecke algebras for $$GL(2)$$. (English) Zbl 0819.11017
Summary: We study the $$p$$-adic nearly ordinary Hecke algebra for cohomological modular forms on $$GL(2)$$ over an arbitrary number field $$F$$. We prove the control theorem and the independence of the Hecke algebra from the weight. Thus the Hecke algebra is finite over the Iwasawa algebra of the maximal split torus and behaves well under specialization with respect to weight and $$p$$-power level. This shows the existence and the uniqueness of the (nearly ordinary) $$p$$-adic analytic family of cohomological Hecke eigenforms parametrized by the algebro-geometric spectrum of the Hecke algebra. As for the size of the algebra, we make a conjecture which predicts the Krull dimension of the Hecke algebra. This conjecture implies the Leopoldt conjecture for $$F$$ and its quadratic extensions containing a $$CM$$ field. We conclude the paper by studying some special cases where the conjecture holds under the hypothesis of the Leopoldt conjecture for $$F$$ and $$p$$.

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F85 $$p$$-adic theory, local fields 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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