Cohen and Strömberg included in their book Modular Forms: A Classical Approach the chapter "A Brief Introduction to Complex Multiplication" (pp. 199-203).
In this chapter (p. 202) we find Proposition 5.10.6, which can be rephrased as follows:
If $\tau$ is a CM point and $\Omega=\eta(\tau)^2$ with the Dedekind $\eta$-function, then $\sqrt{D}E_2^*(\tau)/\Omega^2$ is an algebraic integer, where $D$ is the discriminant of $\tau$.
Unfortunately, the proof begins with "we only prove algebraicity, not the integrality".
Question: How exactly can I prove the integrality? I'm looking for a proof which is at least as detailed as Cohen/Strömberg's book; or a reference for such a proof.
Edit: Here is a direct link to the relevant pages (thanks to reuns): Google Books
Edit 2: The coefficients of the Chudnovsky-Formula can be computed as the value of $$s_2(\tau):=\frac{E_4(\tau)}{E_6(\tau)}\cdot E_2^*(\tau)$$ at $\tau=\frac{1+i\sqrt{163}}{2}$, where we have $$\frac{1-s_2(\tau)}{6} = \frac{13591409}{545140134}.$$ It is easy to compute the approximate value of this, but for proving that this value is exact, I still need to prove integrality. For the precise context, see Prop. 10.2 (page 39), which is used in the end of the proof of 10.3 (page 40).
Also, Berndt and Chan write here on p. 78:
These authors [the Chudnovskys and Borweins] have calculated these coefficients, but we are uncertain if these calculations are theoretically grounded.
Unfortunately, they continue calculating the coefficients using minpoly() in MAPLE. I would like to give a complete proof of the Chudnovsky-Formula, only this bit remains open...