Type III seesaw under $A_4$ modular symmetry with leptogenesis

We make an attempt to study neutrino phenomenology in the framework of type-III seesaw by considering $A_4$ modular symmetry in the super-symmetric context. In addition, we have included a local $U(1)_{B-L}$ symmetry which eventually helps us to avoid certain unwanted terms in the superpotential. Hitherto, the seesaw being type-III, it involves three fermion triplet superfields $\Sigma_R$, along with which, we have included a singlet weighton field $(\rho)$. In here, modular symmetry plays a crucial role by avoiding the usage of excess flavon (weighton) fields. Also, the Yukawa couplings acquire modular forms which are expressed in terms of Dedekind eta function $\eta(\tau)$. However, for numerical analysis we use $q$ expansion expressions of these couplings. Therefore, the model discussed here is triumphant enough to accommodate the observed neutrino oscillation data and also successfully explains observed baryon asymmetry of the universe through leptogenesis.