An elliptic curve over the complex numbers has an associated period lattice, generated by two complex numbers ω1 and ω2, but as well as these two periods there are also two “quasiperiods” η1 and η2 satisfying the Legendre relatin ω1η2 − ω2η1 = 2πi. On the other hand, modular forms of integral weight have associated “period polynomials” whose coefficients are also called periods. In the first part of the lecture I will review the classical construction of these period polynomials (associated with the names Eichler, Shimura, and Manin) and explain that modular forms also have quasiperiods, although this was apparently not noticed until a couple of years ago. In the second part, which can be seen as a continuation of Albrecht Klemm’s talk two weeks ago, we will see that the periods and quasiperiods of certain modular forms appear (conjecturally, and numerically to high precision) in the transition matrices for the Picard-Fuchs differential equations of certain families of Calabi-Yau varieties.
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