Cyclic isogenies and nonstandard arithmetic

For a prime N we denote by X₀(N)(K) the set of K-rational points on the modul curve of elliptic curves with isogenies of degree N. We formulate arithmetical axioms for number fields K that imply finiteness properties of X₀(N)(K). To prove the results we use the nonstandard version of the Siegel-Mahler theorem (A. Robinson and P. Roquette, J. Number Theory7 (1975), 121–176) and the nonstandard interpretation of a sum formula derived from the local heights on elliptic curves.