I read somewhere that said that the modular lambda function, defined as the ratio of two theta functions is algebraic whenever the parameter $\tau$ is defined over a number field, but I cannot find a reference for this.
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$\begingroup$ It is equivalent to saying that the $j$-invariant is algebraic (since $j$ is a rational function of $\lambda$) which should be easier to find references on. But Wikipedia only claims this is true when $\tau$ is imaginary quadratic: en.wikipedia.org/wiki/J-invariant#Class_field_theory_and_j $\endgroup$Qiaochu Yuan– Qiaochu Yuan2024-08-12 17:07:44 +00:00Commented Aug 12, 2024 at 17:07
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$\begingroup$ As written in the previous comment, $\lambda(\tau)$ is algebraic if and only if $j(\tau)$ is algebraic. This is always true if $\tau$ is imaginary quadratic, but if $\tau$ is algebraic but not imaginary quadratic then $j(\tau)$ must be transcendental (see Transcendence properties on en.wikipedia.org/wiki/J-invariant). Also note that $j(\tau)$ can be algebraic for transcendental $\tau$ (in a sense this is the generic case). $\endgroup$user1363745– user13637452024-08-12 18:30:48 +00:00Commented Aug 12, 2024 at 18:30
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