Special values of $j$-invariant

Question

Let $j(\tau)$ be Klein's absolute invariant defined for $\tau \in \mathbb{H}$ by $$j(\tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + \cdots$$ with $q := e^{2\pi i\tau}$. Are there any known special values of $j(\tau)$ for which $\textrm{Re}(\tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $\textrm{Re}(\tau)$ is rational.

Answer 1

The real values of the j-invariant are of interest in theoretical physics (for instance, Witten’s 3d gravity j-invariant is associated to black hole entropy). Now observe that the real number $\phi$, meaning the golden ratio $\phi$, is a special root of the (degree 10) polynomial that one obtains when writing $j$ as $A+iB/ C +iD$ and looking for real parts (using the usual ratio involving the modular discriminant). This is one of few integer values of $27 j$ (watch out for the normalisation factors), like the other special values on the ribbon graph (dessins of Grothendieck).

Why integer values are special is a much more difficult question!

Answer 2

Since it wasn't specified that $\tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(\tau)$ where $\Re(\tau)$ is irrational. However, $\tau$ has a hypergeometric closed-form.

It is known that the equation, $$j(\tau) = n$$ can be solved for $\tau$ in terms of the hypergeometric function. Using Method 4, we find,

$$\tau = \frac{_2F_1\big(\tfrac16,\tfrac56,1,1-\alpha\big)}{_2F_1\big(\tfrac16,\tfrac56,1,\alpha\big)}\sqrt{-1}$$

where

$$\alpha=\frac{1+\sqrt{1-\frac{1728}n}}{2}$$

For example, negating the year $n=-2017$, then,

$$\tau \approx -0.273239 + 0.6868913\,i$$

such that,

$$j(\tau)=-2017$$