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I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite well, except for some inconsistence (from my point of view) in using left/right cosets for actions of the modular group on the set of matrices of determinant $n$. I overcome this trouble by using one more simple arithmetic lemma which relates left and right cosets. I wonder whether there exist another version of the Kronecker--Weber approach for modular polynomials, or maybe even another elementary proof.

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An elementary proof of what? – Pete L. Clark Apr 21 '10 at 2:16
Of the standard properties of modular polynomials which allow one to prove that the algebraicity of values of the modular invariant at CM points. – Wadim Zudilin Apr 21 '10 at 2:34
up vote 4 down vote accepted

I'm not quite sure what exactly you're asking for, but you might find the third part of David Cox's Primes of the form x^2 + n y^2 useful for an elementary approach to modular polynomials.

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Thanks, Alison! I just got the book. It seems that David Cox uses exactly the same approach but takes more care than Lang. – Wadim Zudilin Apr 21 '10 at 2:33
Agreed. This is the most modern reference I know of for the "classical" approach to this material. – Pete L. Clark Apr 21 '10 at 2:45
Although the exposition is slightly tied to considering modular forms for $\Gamma_0(m)$ as well (something I do not cover in my very short course) and is too sketchy at some point, I would agree about its high quality and self-consistence. But Lang's proof (after fixing a minor piece) is simpler. – Wadim Zudilin May 3 '10 at 5:17


I find Silverman's exposition in his "Advanced Topics in the Arithmetic of Elliptic Curves" very clear, and concise. Look at the section on the integrality of the j-invariant, in particular Theorem 6.1 of Chapter II, Section 6, pages 140-151. You can find three proofs of the integrality of the j-invariant and, more concretely, you should have a look at the "Analytic proof of Theorem 6.1", starting in page 143, which is the "classical" one I believe you are interested in (it is also a proof using matrices of determinant n, but the exposition is great).

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It's really a nice and self-contained exposition of the same classical proof, with "disadvantage" of starting with isogenies of elliptic curves. Since in my course the latter appear later, I cannot use it as reference for students. – Wadim Zudilin May 3 '10 at 5:20

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