J. Korean Math. Soc. 2017; 54(4): 1345-1356
Online first article May 24, 2017 Printed July 1, 2017
https://doi.org/10.4134/JKMS.j160518
Copyright © The Korean Mathematical Society.
Junguk Lee
Yonsei University
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.
Keywords: ranks of elliptic curves, nonstandard weak Mordell-Weil property, infinite factorization in nonstandard rationals
MSC numbers: Primary 03H15; Secondary 11G05, 11U10
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