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Probably this is well known.

$\theta_2$ and $\theta_3$ are Jacobi theta functions as defined in mathworld (31) and (32).

For natural $n$ define $$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-e^{-\pi\sqrt{n}}))}$$

According to mathworld (46), $f(3)$ is algebraic.

Experimentally $f(n)$ is algebraic at least for

$$ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 15 , 16 , 18 , 19 , 22 , 23 , 25 , 27 , 28 , 31 , 37 , 39 , 43 , 55 , 58 , 63 , 67 , 163$$

This holds to precision $10^4$ and the list might be incomplete.

The sequence doesn't appear in OEIS though includes all Heegner numbers.

Q1 When is $f(n)$ algebraic?

Q2 Why Heegner numbers are in the list?

Q3 Is there closed form for some other $f(n)$?.

Q4 Is $f(n)$ algebraic for other $n$?

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    $\begingroup$ $f(n)$ is essentially the "lambda function" (mathworld.wolfram.com/EllipticLambdaFunction.html) or the "elliptic modulus", and I think the indices correspond (maybe not 1-to-1) to "elliptic integral singular values" (mathworld.wolfram.com/EllipticIntegralSingularValue.html) $\endgroup$
    – Fredrik Johansson
    Commented Nov 4, 2014 at 16:32
  • $\begingroup$ @FredrikJohansson Thank you :-). Why not 1-to-1? $\endgroup$
    – joro
    Commented Nov 4, 2014 at 16:46
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    $\begingroup$ They're all algebraic, by the theory of complex multiplication. For example, for $\ n=14$ (the first number not in your list) I find that $2f(n)^4$ is a root of $$ x^8 - 8x^7 - 1964x^6 + 11896x^5 - 27034x^4 + 28680x^3 - 14764x^2 + 3976x + 1. $$ Values at "Heegner numbers" are easiest to recognize because the degree is small. $\endgroup$
    – Noam D. Elkies
    Commented Nov 4, 2014 at 16:54
  • $\begingroup$ @NoamD.Elkies is the degree of $f(n)^4$ bounded (possibly conjecturally) infinitely often? $\endgroup$
    – joro
    Commented Nov 5, 2014 at 17:08
  • $\begingroup$ On the contrary, it's known that the degree tends to $\infty$ as $n \rightarrow \infty$, because it's within a bounded factor of the class number of ${\bf Q}(\sqrt{-n}\,)$. $\endgroup$
    – Noam D. Elkies
    Commented Nov 6, 2014 at 1:16

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