This thesis consists of two parts. Both parts are about the algebra of theta functions. The first part is on the construction of the theta Hecke algebra, which contains both the graded algebra of theta functions and the Hecke algebra of the Heisenberg group with respect to certain discrete subgroup. The idea is originated from the theory of Connes-Moscovici. They construct the modular Hecke algebra which contains the usual graded algebra of modular forms for T, a congrunce subgroup of SL(2,Z), and the Hecke algebra of (GL⁺(2,Q),T) as subalgebras. In this paper, I will determine the structure of the theta Hecke algebra and express the multiplication explicitly....[ Read more ]

This thesis consists of two parts. Both parts are about the algebra of theta functions. The first part is on the construction of the theta Hecke algebra, which contains both the graded algebra of theta functions and the Hecke algebra of the Heisenberg group with respect to certain discrete subgroup. The idea is originated from the theory of Connes-Moscovici. They construct the modular Hecke algebra which contains the usual graded algebra of modular forms for T, a congrunce subgroup of SL(2,Z), and the Hecke algebra of (GL⁺(2,Q),T) as subalgebras. In this paper, I will determine the structure of the theta Hecke algebra and express the multiplication explicitly.

The second part is devoted to discuss certain deformations of the algebra of theta functions. The theta functions can be described as the global sections of the basic line bundles of an abelian variety. On the other hand, it can be realized as the invariant subspace of certain representation of the Heisenberg group by certain discrete subgroup. I will first give two examples of deformations of the algebra of theta functions. The first one uses the method by Artin-Tate-Van Den Bergh. The other one is new, and it uses the Heisenberg group representation. Then I will propose a general construction by the use of Heisenberg group representation and show that the above two examples are the special case of this construction. We know that the algebra of theta functions determines the structure of the underlying abelian varieties. A non-commutative deformation of the algebra may be interpreted as a deformation of an abelian variety to some geometric objects in non-commutative geometry.