Abelian modular symbols over real quadratic fields

Abelian modular symbol (over $\mathbb Q$) is first introduced by Hida in his blue book to reformulate the construction of the Kubota-Leopoldt $p$-adic $L$-function. I have shown a certain homological distribution result for the symbols to reprove residual non-vanishing result for special Dirichlet $L$-values with cyclotomic twists, namely Washington's Theorem. In the talk, I will discuss how to construct the symbols over real quadratic fields and present a possible approach to study the residual non-vanishing problem for special Hecke $L$-values of real quadratic fields with cyclotomic twists, which is a generalization of my previous argument for $\mathbb Q$. This is ongoing research and is joint work with Jungyun Lee and Jaesung Kwon.