Sp(6, Z) modular symmetry in flavor structures: quark flavor models and Siegel modular forms for $$\widetilde{\Delta }\left(96\right)

We study an approach to construct Siegel modular forms from Sp(6, Z ). Zero-mode wave functions on T 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 the S -symmetry remain. Consequently, we obtain Siegel modular forms with three moduli parameters ( ω 1 , ω 2 , ω 3 ), which are multiplets of finite modular groups. We show several examples. As one of examples, we study Siegel modular forms for $$\widetilde{\Delta }\left(96\right)$$ in detail. Then, as a phenomenological applicantion, we study quark flavor models using Siegel modular forms for $$\widetilde{\Delta }\left(96\right)$$ . Around the cusp, ω 1 = i∞ , the Siegel modular forms have hierarchical values depending on their T I -charges. We show the deviation of ω 1 from the cusp can generate large quark mass hierarchies without fine-tuning. Furthermore CP violation is induced by deviation of ω 2 from imaginary axis.