SOME ARITHMETIC PROPERTIES ON NONSTANDARD NUMBER FIELDS

作者：Lee Junguk^*

来源：Journal of the Korean Mathematical Society, 2017, 54(4): 1345-1356.

DOI：10.4134/JKMS.j160518

• ranks of elliptic curves
• nonstandard weak Mordell-Weil property
• infinite factorization in nonstandard rationals

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 *Z-module, where *Z is an ultrapower of Z, 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 Z, which is definable. We can consider definable abelian groups as *Z-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.

• 出版日期2017-7