P-ADIC ORDINARY HECKE ALGEBRAS FOR GL(2)

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 alebro-geometric spectrum of the Hecke algebra. As for a 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 studying some special cases where the conjecture holds under the hypothesis of the Leopoldt conjecture for F and p.