Abstract
For a prime N we denote by X 0( 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 X 0( N)( K). To prove the results we use the nonstandard version of the Siegel-Mahler theorem ( A. Robinson and P. Roquette, J. Number Theory 7 (1975) , 121–176) and the nonstandard interpretation of a sum formula derived from the local heights on elliptic curves.
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