Finiteness of an Isogeny Class via Equidistribution

In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings’ isogeny theorem over number fields by reducing to Tate’s theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.