Introduction to Arithmetic Mirror Symmetry

Abstract

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.

Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

“…We list the corresponding maps in Table 5; as they are not uniquesome instances differ from the descriptions in other sources, for example, in [12]. In the literature, there are many papers on counting the number of solutions of these algebraic equations of type (37) over finite fields; see, e.g., [18,26,27,45,55]. In particular, it is shown in [55] that V α (ψ) and its mirror V α (λ) share the same unit root.…”