Quaternionic big Heegner points over totally real fields

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach simultaneously treats totally definite and indefinite quaternion algebras. We then extend their interpolation methods to define big Heegner points attached to families of Hilbert modular forms of parallel weight under the weak Heegner hypothesis. Applying this construction, we build in the definite setting a totally real analogue of Longo$-$Vigni's two-variable $p$-adic $L$-function, and in the indefinite setting, a system of big Heegner classes in the sense of Howard.