Arithmetic Properties of Moduli Spaces and Topological String Partition Functions of Some Calabi-Yau Threefolds

This thesis studies certain aspects of the global properties, including geometric and arithmetic, of the moduli spaces of complex structures of some special Calabi-Yau threefolds (B-model), and of the corresponding topological string partition functions defined from them which are closely related to the generating functions of Gromov-Witten invariants of their mirror Calabi-Yau threefolds (A-model) by the mirror symmetry conjecture.;For the mirror families (B-model) of the one-parameter families (A-model) of KP2, KdPn, n = 5,6,7,8 with varying Kahler structures, the bases are the moduli spaces of complex structures of the corresponding mirror Calabi-Yaus. We identify them with certain modular curves by studying the Picard-Fuchs systems and periods of the corresponding mirror families. In particular, the singular points on the moduli spaces correspond to the cusps and elliptic points on the modular curves.;We take the BCOV holomorphic anomaly equations with boundary conditions as the defining equations for the topological string partition functions. Using polynomial recursion and the above identification, we interpret the boundary conditions as regularity conditions for modular forms and express the equations purely in terms of the language of modular form theory. This turns the problem of solving the equations into a combinatorial problem. We also solve for the first few topological string partition functions genus by genus recursively in terms of almost-holomorphic modular forms. Assuming the validity of mirror symmetry conjecture, we prove a version of integrality for the Gromov-Witten invariants of the original non-compact Calabi-Yau threefolds (A-model) as a consequence of the modularity of the partition functions.;Motivated by the results for the aforementioned non-compact Calabi-Yaus, we construct triples of differential rings on the moduli spaces of complex structures for some one-parameter families of compact Calabi-Yau threefolds (B-model), e.g., the quintic mirror family, in a systematic way. These rings are defined from the Picard-Fuchs equations and special Kahler geometry on the moduli spaces. They share structures similar to the triples of rings of modular forms, quasi-modular forms and almost-holomorphic modular forms defined on modular curves. Moreover, the topological string partition functions are Laurent polynomials in the generators of the differential rings.