Abelian modular symbols over real quadratic fields

Abstract

Abelian modular symbol (over 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 Q. This is ongoing research and is joint work with Jungyun Lee and Jaesung Kwon.