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Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $ X_0(N)$ because the function field of $X_0(N)$ is $\mathbb C(j(Nz),j(z))$ and $\Phi_N(j(Nz),j(z))=0$.

Suppose that $\tau$ is a CM point, and let $\mathcal O$ be the corresponding quadratic order. In this case we can use the modular equation to obtain a first proof that $j(\tau)$ is algebraic - we just find a matrix fixing the point $\tau$. In particular, if the class number of $\mathcal O$ is one, then $j(\tau)$ is rational.

A second approach to the rationality of the $j$-invariant is as follows. Let $E$ be an elliptic curve such that $\operatorname{End}(E)\cong \mathcal O$. Then $j(E)=j(\tau), \operatorname{End}(E)\cong\operatorname{End}(E^\sigma)$ and $j(E)^\sigma= j(E^\sigma)$ for any automorphism $\sigma$. But there are exactly $h(\mathcal O)$ nonisomorphic elliptic curves with CM by $\mathcal O$, hence the degree of $j(\tau)$ is at most $h(\mathcal O)$.

Using the $j$-invariant we can view isomorphism classes of elliptic curves as points in $\mathbb C$. In other words, elliptic curves correspond to noncuspidal points of $X(1)$. The foregoing proof therefore shows that quadratic orders with class number one give rise to rational points on $X(1)$. Since $X(1)$ has genus zero, this gives us no useful information. If, however, we choose other suitable modular curves with higher genus, we can determine all rational points on them, and consequently all imaginary quadratic fields with class number one (as observed by J. P. Serre).

I would like to ask the following:

  1. Are the two proofs mentioned really different?
  2. Is there a geometric interpretation of the first proof?
  3. What role does the curve $X_0(N)$ really play in the first proof?
  4. Can we formulate the first proof using only the language of moduli spaces and isogenies?

References

B. Baran: Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

J. H. Silverman: Advanced Topics in the Arithmetic of Elliptic Curves - second chapter

F. Diamond, J. Shurman: A First Course in Modular Forms - first and seventh chapter

Please keep the discussion elementary - say, at the level of the above references.

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