On the height of the universal abelian variety

Abstract: In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept used in this extension is the relative finite energy introduced by T. Darvas, E. Di Nezza, and C. H. Lu. As an application, we prove that the line bundle of Siegel--Jacobi forms on the universal abelian variety endowed with its invariant hermitian metric is not integrable but we compute its arithmetic self-intersection number using the new extension. The techniques developed in this paper can be applied in many other situations like mixed Shimura varieties or the moduli space of stable marked curves.