A height bound for abelian schemes with real×Q2 multiplication

In this paper, we prove a height bound for points on the base of a family of abelian varieties at which the fibre possesses additional endomorphisms. This complements a result of André in his book (G-Functions and Geometry Aspects of Mathematics, E13. Friedrich Vieweg and Sohn, Braunschweig, 1989) as well a result of Daw and Orr (Ann Scuol Norm Super Class Sci 39:1, 2021). The work in this paper will be used to prove a new case of the Zilber-Pink conjecture which will form part of the author’s PhD thesis