The modified diagonal on the triple product of a curve was first introduced by Gross and Schoen in the 90’s. This simply defined object holds fundamental importance in the study of the geometry and arithmetic of curves. One basic question is whether the modified diagonal vanishes under “deformation”. I will introduce the origin of this type of question and provide a brief history of the study of the modified diagonal. Subsequently, I will discuss my collaborative works with W. Zhang, where we demonstrated that such vanishing can be dictated by the symmetry of the curve. As an application in number theory, we proved a case of the notorious Beilinson—Bloch conjecture, a generalization of the Millennium Birch—Swinnerton-Dyer conjecture. Finally, I want to propose some new questions.