Some arithmetic properties on nonstandard rationals

TL;DR

This paper explores the properties of nonstandard rational numbers and their elliptic curves, establishing equivalences between various Mordell-Weil properties in nonstandard models and analyzing prime ideals using valuation theory.

Contribution

It introduces the Nonstandard Mordell-Weil property and proves its equivalence to the weak Mordell-Weil property in nonstandard rational fields, extending classical number theory concepts.

Findings

Equivalence of Nonstandard Mordell-Weil and weak Mordell-Weil properties.

Infinite factorization theorem for nonstandard rationals using valuations.

Classification of prime ideals in nonstandard rational number fields.

Abstract

For a given number field $K$, we show that the ranks of nonsingular elliptic curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce Nonstandard Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*Z$-module, where $^*Z$ is an ultrapower of $Z$, and we show that Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property in $^*K$. In Appendix, we showed that it is possible to consider definable abelian groups as $^*Z$-modules in a saturated nonstandard rational number field $^*Q$ so that nonstandard Mordell-Weil property is well-defined, and thus we showed that nonstandard Mordell-Weil property and weak Mordell-Weil property are equivalent. Next we focus on priems and prime ideals of nonstandard raional number fields. We give an infinite factorization theorem on $^*Q$ using valuations induced from primes of $^*Z$, and we classify maximal and prime ideal of $^*Z$ in terms of maximal filter on the set of primes of $^*Z$ and ordered semigroups of the valuation semigroup induced from maximal ideals of $^*Z$.