SOME ARITHMETIC PROPERTIES ON NONSTANDARD NUMBER FIELDS

For a given number field $K$, we show that the ranks of elliptic curves over $K$ are uniformly finitely bounded if and only if the weak Mordell-Weil property holds in all (some) ultrapowers $^*K$ of $K$. We introduce the nonstandard weak Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*\BZ$-module, where $^*\BZ$ is an ultrapower of $\BZ$, and we show that the nonstandard weak Mordell-Weil property is equivalent to the weak Mordell-Weil property in $^*K$. In a saturated nonstandard number field, there is a nonstandard ring of integers $^*\BZ$, which is definable. We can consider definable abelian groups as $^*\BZ$-modules so that the nonstandard weak Mordell-Weil property is well-defined, and we conclude that the nonstandard weak Mordell-Weil property and the weak Mordell-Weil property are equivalent. We have valuations induced from prime numbers in nonstandard rational number fields, and using these valuations, we identify two nonstandard rational numbers. KCI Citation Count: 0