ARITHMETIC PROPERTIES OF MIRROR MAP AND QUANTUM COUPLING

We study some arithmetic properties of the mirror maps and the quantum Yukawa couplings for some 1-parameter deformations of Calabi-Yau mani folds. First we use the Schwarzian differential equation, which we der ived previously, to characterize the mirror map in each case. For alge braic K3 surfaces, we solve the equation in terms of the J-function. B y deriving explicit modular relations we prove that some K3 mirror map s are algebraic over the genus zero function field Q(J). This leads to a uniform proof that those mirror maps have integral Fourier coeffici ents. Regarding the maps as Riemann mappings, we prove that they are g enus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we f ind that these maps are actually the reciprocals of the Thompson serie s for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced mod p. Under th e mirror hypothesis and an integrality assumption, we derive mod p con gruences for the Fourier coefficients. For the quintics, we deduce, (a t least for 5 Xd) that the degree d instanton numbers n(d) are divisib le by 5(3) - a fact first conjectured by Clemens.