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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.

L. Milla
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  • 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. – Dietrich Burde May 24 '18 at 15:23
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    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 – Mathmo123 May 24 '18 at 15:59
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    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 – User0112358 May 25 '18 at 07:16

2 Answers2

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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".

David Loeffler
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Standard introductory texts are:

  • Rational Points on Elliptic Curves - Silverman and Tate

  • Elliptic Curves - Milne

  • Introduction to Elliptic Curves and Modular Forms - Koblitz

More advanced texts are:

  • The Arithmetic of Elliptic Curves - Silverman

  • Advanced Topics in the Arithmetic of Elliptic Curves - Silverman

gandalf61
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    Given that at least 3 of these books barely touch on complex multiplication, I'm not really sure what this answer gives. – Mathmo123 May 24 '18 at 15:48
  • The questioner asks about modular forms as well as complex multiplication. But by all means feel free to improve on my suggestions :) – gandalf61 May 24 '18 at 18:10
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    Only one of these books contains anything substantial about modular forms. These are 5 serious texts, of which only the last is useful for the problem at hand. The OP could easily waste a lot of time obtaining and reading through each of the first four. On what basis did you recommend these texts? Have you read them? – Mathmo123 May 24 '18 at 18:17