The author begins a study of the theory of p-adic Hecke algebras for \(GL_ 2\) over a totally real field F. As part of this study he generalizes and extends many of his earlier results in the case when \(F={\mathbb{Q}}\). However, in the general case, the existence of multiple weight modular forms means that there are infinitely many distinct Hecke algebras parametrized by the classes of weights modulo parallel ones (a weight \((k_ 1,...,k_ d)\) is parallel if \(k_ i=k_ j\) for all i and j). The isomorphism between the ordinary Hecke algebras of different weights k and \(k'\) occurs only when k and \(k'\) are parallel.
A major part of the paper is devoted to studying the Hecke algebra as a module over what is essentially the continuous group algebra A (respectively, \(\Lambda)\) of the Galois group Z (respectively, the torsion-free part of Z) of the maximal abelian extension of F which is unramified away from the level and \(\infty\). For example, the ordinary Hecke algebra is shown to be a torsion-free \(\Lambda\)-module of finite type. The author also proves duality theorems between Hecke algebras and spaces of cusp forms, and derives consequences for p-adic modular forms. The proofs depend upon the theory of p-adic Hilbert modular forms, and the study of the \(\Lambda\)-module structure of the cohomology of arithmetic subgroups of quaternion algebras over F.
It should be mentioned that one of the motivating forces behind this study is the possibility of constructing Galois representations over the Hecke algebra. In certain cases this has been done by A. Wiles [Invent. Math. 94, 529-573 (1988)] using results of the author.