Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for (△)over-tilde(96)

We study an approach to construct Siegel modular forms from Sp(6, Z). Zero-mode wave functions onT(6) with magnetic flux background behave Siegel modular forms at the origin. Then T-symmetries partially break depending on the form of background magnetic flux. We study the background such that three T-symmetries T-I, T-II and T-III as well as theS-symmetry remain. Consequently, we obtain Siegel modular forms with three moduli parameters(omega(1), omega(2), omega(3)), which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for (triangle) over tilde (96)in detail. Then, as aphenomenological applicantion, we study quark flavor models using Siegel modular forms for (triangle) over tilde (96). Around the cusp, omega(1)= i infinity, the Siegel modular forms have hierarchical values depending on their T-I-charges. We show the deviation of omega(1) from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of omega(2) from imaginary axis.