The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called 'Gauss-Manin connection in disguise'.

Hodge theory—one of the pillars of modern algebraic geometry—is a deep theory with many applications and open problems, the most enigmatic of which is the Hodge conjecture, one of the Clay Institute's seven Millennium Prize Problems. Hodge theory is also famously difficult to learn, requiring training in many different branches of mathematics. The present volume begins with an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann, among many others; and then the studies of two-dimensional multiple integrals by Poincaré and Picard. Thenceforth, the focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials, for which many tools from singularity theory, such as Brieskorn modules and Milnor fibrations, are used. Another aspect of this volume is its computational presentation of many well-known theoretical concepts, such as the Gauss–Manin connection, homology of varieties in terms of vanishing cycles, Hodge cycles, Noether–Lefschetz, and Hodge loci. All are explained for the generalized Fermat variety, which for Hodge theory boils down to a heavy linear algebra. Most of the algorithms introduced here are implemented in Singular, a computer algebra system for polynomial computations. Finally, the author introduces a few problems, mainly for talented undergraduate students, which can be understood with a basic knowledge of linear algebra. The origins of these problems may be seen in the discussions of advanced topics presented throughout this volume.

This book intends to construct a theory of modular forms for families of Calabi-Yau threefolds with Hodge numbers of the third cohomology equal to one. It discusses many differences and similarities between the new theory and the classical theory of modular forms defined on the upper half plane. The main examples of the new theory are topological string partition functions which encode the Gromov-Witten invariants of the mirror Calabi-Yau threefolds. It is mainly written for two primary target audiences: researchers in classical modular and automorphic forms who wish to understand the q-expansions of physicists derived from Calabi-Yau threefolds, and mathematicians in enumerative algebraic geometry who want to understand how mirror symmetry counts rational curves in compact Calabi-Yau threefolds. This book is also recommended for mathematicians who work with automorphic forms and their role in algebraic geometry, in particular for those who have noticed that the class of algebraic varieties involved in their study is limited: for instance, it does not include compact non-rigid Calabi-Yau threefolds. A basic knowledge of complex analysis, differential equations, algebraic topology and algebraic geometry is required for a smooth reading of the book.