Research PaperNo Access

Diversity in rationally parameterized number fields

Benjamin Klahn

Institute of Analysis and Number Theory, Technical University of Graz, Ḱopernikusgasse 24/II, Graz, Austria

https://doi.org/10.1142/S1793042123501154Cited by:0 (Source: Crossref)

Abstract

Let $X$ be a curve defined over $ℚ$ and let $t \in ℚ (X)$ be a non-constant rational function on $X$ of degree $v \geq 2$ . For every rational number $a / b$ pick a point $P_{a / b} \in X (\bar{ℚ})$ such that $t (P_{a / b}) = a / b$ . In this paper, we obtain lower bounds on the number of distinct fields among $ℚ (P_{a / b})$ with $1 \leq a, b \leq N$ under some assumptions on $t$ . We show that if $t$ has a pole of order at least 2 or if there is a rational number $α$ such that $t - α$ has a zero of order at least 2, then the set ${ℚ (P_{a / b}) | 1 \leq a, b \leq N}$ contains $≫ \frac{N^{2}}{{(log N)}^{2}}$ elements. We also obtain partial results when $t$ does not have a pole of order at least two.

Keywords:

AMSC: 11C08, 11G99, 11R09

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