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LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 12 September 2018, pp. 42-50
- Print publication:
- February 2019
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- Article
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We show that two distinct singular moduli
$j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers 
$m$ and 
$n$ the numbers 
$1,j(\unicode[STIX]{x1D70F})^{m}$ and 
$j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over 
$\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.
