One of the main issues in modern theory of automorphic forms is to understand the structure of square integrable automorphic forms. The main approach is to use Arthur- Selberg trace formula and its stabilization. From this the endoscopy structures of square integrable automorphic forms was discovered. By the recent work of B.-C. Ngo on the Fundamental Lemma, the existence of such endoscopy structures are expected to be established by Arthur based on the work of Waldspurger, and of Laumon and others.

In this talk, I will discuss explicit constructions of certain families of endoscopy transfers in terms of integral operators with certain residues of Eisenstein series as the kernel functions. This is mainly the on-going project of mine with David Ginzburg and David Soudry.