The flavor structures on magnetized orbifold models and 4D modular symmetric models

We study quark and lepton flavor structures on magnetized $T^2/\mathbb{Z}_2$ twisted orbifold model. There are 6,460 number of flavor models but most of them cannot lead to realistic flavor observables because of the difficulties on realizing mass hierarchies and small (large) mixing angles of quarks (leptons). We find that certain zero point patterns of zero-modes of fermions and Higgs fields give the flavor models being able to avoid these difficulties. We classify such flavor models and show numerical example. Next we study four-dimensional (4D) modular symmetric quark flavor models without fine-tuning. Mass matrices become hierarchical as close to the modular symmetric points depending on its residual charges. Actually the residual $Z_N$ symmetries with $N\geq 6$ can originate the large quark mass hierarchies. Also the products of residual symmetries such as $Z_3\times Z_3\times Z_3$ have such possibility. We study the quark flavor model with $\Gamma_6$ symmetry and $A_4\times A_4\times A_4$ symmetry. We assume the vicinity of the cusp $\tau=i\infty$ where residual $Z_6$ and $Z_3\times Z_3\times Z_3$ symmetry remain. Consequently we find the models realizing the order of the quark mass ratios and the absolute values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements without fine-tuning. Finally we construct the Siegel modular forms. Zero-modes on $T^6$ at $\vec{z}=0$ are the Siegel modular forms of weight 1/2 for the subgroup of $Sp(6,\mathbb{Z})$. They have several moduli parameters and therefore have the possibility realizing the flavor structures including the CP phases. We study the Siegel modular forms transformed by $\widetilde{\Delta}(96)$ and show numerical example. We find one of moduli parameters $\omega_1$ works on the large mass hierarchies and $\omega_2$ works on the CP violation successfully in our model.