I was reading a set of notes by Zijian Yao titled "Mordell's conjecture after Faltings and Lawrence-Venkatesh". I am unable to find a source or the proof of the following statement which appears as Proposition 3.5 in the aforementioned set of notes (Here is the link to the notes http://people.math.harvard.edu/~zyao/notes/notes/mordell.pdf)
Proposition : Let $K$ be a number field. Fix $A_0$ an abelian variety over $K$ and an integer $d \geq 1$, then upto isomorphism there are only finitely many polarized abelian varieties $(A, \lambda)$, where $A \cong A_0$ and $\lambda$ has degree $d$.
Any hint of the proof or references are most welcome. Thanks!
The desired statement follows from Theorem 1.1 in [1] for algebraically closed fields. To obtain the result for a number field we may use finiteness of number of forms of an abelian variety which follows from Theorem 6.1 in [2].
[1] Narasimhan, M.S., Nori, M.V. Polarisations on an abelian variety. Proc. Indian Acad. Sci. (Math. Sci.) 90, 125–128 (1981). https://doi.org/10.1007/BF02837283
[2] Borel, A., Serre, J.P. Théorèmes de finitude en cohomologie galoisienne. Commentarii Mathematici Helvetici 39, 111–164 (1964). https://doi.org/10.1007/BF02566948