pro-Heisenberg modules and real multiplication

Noncommutative tori can be viewed as limits of elliptic curves for which the period lattice degenerates to a pseudo-lattice (a rank-2 free subgroup of the real line). A noncommutative torus whose period pseudo-lattice correspond to an order in a real quadratic field is called a real multiplication noncommutative torus. Based on strong analogies with the case of elliptic curves with complex multiplication Y. Manin has pointed out the possible relevance of this class of noncommutative tori in providing the right geometric framework underlying explicit class field theory of real quadratic fields. In this talk we discus a class of modules arising naturally in this context from towers of self-isogenies of real multiplication noncommutative tori. Self isogenies in this context are given in terms of representations of Heisenberg groups implementing morita self-equivalences of the underlying coordinate rings. The modules we construct play a role analogous to representations of adelic Heisenberg groups arising from ample line bundles on elliptic curves. The corresponding algebraic theta functions functions and their relation to the real multiplication program outlined above will also be discussed.