Introduction to Arithmetic Mirror Symmetry
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.
The author would like to thank the anonymous referee for helpful remarks that resulted in large improvements to this document. Thanks is also due to Professor Noriko Yui for helpful suggestions and tireless encouragement during the preparation of this manuscript. The author’s work is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada through the Discovery Grant of Noriko Yui. The author held a visiting position at the Fields Institute during the preparation of these notes, and would like to thank this institution for its hospitality.
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