arXiv subjects

Antonin Riffaut

Publications and source records attributed to Antonin Riffaut.

3 recordsLinked to original sources

arXiv2018

Equations with powers of singular moduli

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ),j(τ') such that the numbers 1, j(τ)^m and j(τ')^n are linearly dependent over $\mathbb{Q}$ for some positive integers m,n, must be of degree at most 2. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number.

math.NT↗

arXiv2018

Fields generated by sums and products of singular moduli

We show that the field $\mathbb{Q}(x,y)$, generated by two singular moduli~$x$ and~$y$, is generated by their sum ${x+y}$, unless~$x$ and~$y$ are conjugate over~$\mathbb{Q}$, in which case ${x+y}$ generates a subfield of degree at most~$2$. We obtain a similar result for the product of two singular moduli.

math.NT↗

arXiv2017

Linear independence of powers of singular moduli of degree 3

We show that two distinct singular moduli $j(τ),j(τ')$, such that for some positive integers $m, n$ the numbers $1,j(τ)^m$ and $j(τ')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree $3$. The purpose of this article is to treat these two remaining cases.

math.NT↗