2000/12/24 by Philip Candelas, Xenia de la Ossa, Candelas, Philip +3 · 1 citation
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0012233
75 pages, 6 eps figures
arxiv created 2000/12/24 · arxiv updated 2009/11/30
We study Calabi-Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show also, for a one parameter family of quintic threefolds, that the number of rational points of the manifold is closely related to as the number of rational points of the mirror manifold. Our interest is primarily with Calabi-Yau threefolds however we consider also the interesting case of elliptic curves and even the case of a quadric in CP1 which is a zero dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial dependence on the parameter over C but a not trivial arithmetic structure.