4

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$

Question: Given a real $x$, $0\lt x\lt 1$, is there an algorithm which determines whether $f(x)$ is positive rational?

Thoughts: Let $\lambda^*(x)=\sqrt{\lambda (i\sqrt{x})}$ where $\lambda$ is the modular lambda function: $$\lambda (x)=\left(\frac{\sum_{n=-\infty}^\infty e^{\pi i x\left(n+\frac{1}{2}\right)^2}}{\sum_{n=-\infty}^\infty e^{\pi ixn^2}}\right)^4$$ It turns out that $\lambda^*$ is the inverse of $f$: $$\lambda^*(f(x))=x$$

It is known that $\lambda^*(f(x))$ is algebraic if $f(x)$ is positive rational. Equivalently, $x$ is algebraic if $f(x)$ is positive rational. By contrapositive,

If $x$ is not algebraic, then $f(x)$ is not positive rational.

So, a part of the problem is already solved. It remains to solve

If $x$ is algebraic, can we determine whether $f(x)$ is positive rational?

The answer is known for some algebraic $x$, for example $$f\left((\sqrt{10}-3)(\sqrt{2}+1)^2\right)=\frac{2}{5}.$$

Nomas
  • 417
  • 1
    for a rational number $r > 0$, the degree of the algebraic number $\lambda^(r)$ is related (up to a uniformly bounded factor) to the order of a certain ray class group. In particular, because of the (effective) solution to the class number problem, you can (in principle) determine all $r$ such that $\lambda^(r)$ has degree at most $N$ for any $N$. But this gives an algorithm, because given an algebraic number $x$ of degree $N$ you just enumerate all $x$ and then check each one. – user952367 Sep 10 '21 at 21:39
  • @user952367 Can you please provide more detail or give an example? For example for $f\left(\frac{\sqrt{2}}{7}\right)$. – Nomas Sep 10 '21 at 22:46
  • "you can (in principle) determine all $r$ such that $\lambda^*(r)$ has degree at most $N$ for any $N$." How would that be done? Where can I find this claim? Do you have some references? – Nomas Sep 11 '21 at 10:08
  • Class field theory is a big subject. If you want to explore it further you are free to do so on your own, that's why this is left in a comment. – user952367 Sep 11 '21 at 12:04
  • 1
    @user952367 Class field theory is a too broad term. Can you please refer me to some specific book which discusses the idea that "you can (in principle) determine all $r$ such that $\lambda^*(r)$ has degree at most $N$ for any $N$."? – Nomas Sep 11 '21 at 18:40
  • @Nomas. Are there any properties or identities known for $\lambda^*(x)$? – Rounak Sarkar Sep 20 '21 at 08:03

0 Answers0