# Good Books on Complex Multiplication (on Elliptic Curves)

I don't know anything yet about Complex Multiplication (on elliptic Curves). Do you know good introductory books on the Matter? I'm especially looking for a proof that the following terms are algebraic integers whenever $\tau$ is a quadratic irrationality: $$\frac{E_4(\tau)}{\eta^8(\tau)},\quad\frac{E_6(\tau)}{\eta^{12}(\tau)}\quad\text{and}\quad\sqrt{D}\cdot\frac{E_2(\tau)-\frac{3}{\pi\Im(\tau)}}{\eta^4(\tau)}$$ Here $\eta$ denotes the Dedekind $\eta$-Function, $E_k$ are the Eisenstein series of weight $k$, and $D=B^2-4AC$ where $A\tau^2+B\tau+C=0$

Edit: As @Mathmo123 commented (Thank you!!!), the first expression is a root of the polynomial $P(X)=X^3-1728J(\tau)$ and the second is a root of $Q(X)=X^2-1728+1728J(\tau)$, where $1728J(\tau)\in\mathbb Z$. Thus it only remains to prove that the third expression is an algebraic integer.

Edit: I verified numerically that the sixth power of the third expression is integral for all Discriminants with class number $1$, but I don't see how I can prove it. Unfortunately, I couldn't find anything in the books mentioned by @gandalf61 or by @Mathmo123 ... -> but this is another topic, moved to another question here.

• Just check good books on elliptic curves. There are many references given at this site. There are also posts about what makes complex multiplication so interesting, see here. May 24, 2018 at 15:23
• I really enjoyed these lecture notes by Ben Green. I'm not sure if they cover the exact forms in your question, but they do cover the integrality of the $j$-invariant in detail. (Your first two expressions are just its square and cubed root.) Alternatively, there is Silverman's Advanced Topics May 24, 2018 at 15:59
• David Cox' "Primes of the form $p = x^2 + ny^{2}$" has a nice section on complex multiplication. However, I can't remember if it has precisely the results you're looking for. Nevertheless, it was a good place for me to start learning about CM May 25, 2018 at 7:16

The chapter on CM theory in Silverman's "Advanced Topics", mentioned in @gandalf61's answer, is a standard introductory account. However, given your recent questions, I suspect you are looking for a textbook with a more complex-analytic flavour, emphasising links with elliptic-function theory, reciprocity laws for singular moduli, etc. In that case, you might like to have a look at Reinhard Schertz's 2010 book "Complex Multiplication".

Standard introductory texts are:

• Rational Points on Elliptic Curves - Silverman and Tate

• Elliptic Curves - Milne

• Introduction to Elliptic Curves and Modular Forms - Koblitz