Families of Heegner points in the totally definite quaternionic setting.

In the same way that one can consider Heegner points on elliptic curves or on weight-two modular forms, one may extend this construction to fit into a Hida family. These objects, first studied by B. Howard in 2007, are known as ``big Heegner points'' because, by specializing them, one recovers the classical Heegner points of V. Kolyvagin. In this short talk, I will sketch a work in progress aimed at extending these ideas into a general quaternionic setting over a totally real field. Although this method, based on optimal embeddings, treats both the indefinite and the totally definite quaternionic settings simultaneously, in this talk we will restrict ourselves to the latter, which has remained largely unexplored in the literature until now. This approach, using towers of ``Gross curves’’, generalizes the work of M. Longo and S. Vigni (2011) to the totally real setting.