Calabi-Yau Threefolds, Counting Points and Physics

For families of Calabi-Yau threefolds, we derive an explicit formula to count the number of points over F_q in terms of the periods of the holomorphic three-form, illustrated by the one-parameter mirror quintic and the 5-parameter Hulek-Verrill family. The formula holds for conifold singularities and naturally incorporates p-adic zeta values, the Yukawa coupling and modularity in the local zeta function. I will give a brief introduction on the physics motivation and how this framework links arithmetic, geometric and physics.