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.
General
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