Ramanujan's constant, $e^{\pi\sqrt{163}}$, is almost an integer. I know that this comes from the Laurent series for the $j$ function in terms of $e^{2\pi\tau}$: $$j(\tau)=q^{-1}+744+196884q+\dots$$ But I do not know exactly why this is. First of all, the $j$-function is defined in a horribly complicated and unintuitive way. Why is it even a modular form (or function, but I don't even know the difference)? More importantly, why is $j(\frac{-1+\sqrt{163}}2)$ an integer, and how does this relate to the class number of $Q(\sqrt{-163})$ being $1$? I have read that it has to do with similar quadratic forms, but how does this relate to the class number or the $j$-function?
EDIT: I realized I have no hope of understanding what is going on with proving that the outputs of the $j$-function to similar quadratic forms being conjugate. Thus, I instead ask:
- Why is the $j$-function modular? What's the difference between a modular form and a modular function?
- Why is the class number equal to the number of distinct similar quadratic forms?
These things are written in Silverman's book 'Advanced topics in the arithmetic of elliptic curves', chapter $2$, complex multiplication.
Let $E$ be an elliptic curve over $\Bbb{C}$, which has CM by $\Bbb{Q}(\sqrt{-163}$) (because class number of $\Bbb{Q}(\sqrt{-163})$, there is only one elliptic curve which has CM by $\Bbb{Q}(\sqrt{-163}$ up to isomorphism). Let $τ=\frac{1＋\sqrt{-163}}{2}$.
Because j-invariant of CM elliptic curve is algebraic integer and $j(τ)\in \Bbb{Q}$(Here we used class number of $\Bbb{Q}(\sqrt{-163})$ is one, because $[\Bbb{Q}(j(E)):\Bbb{Q}]=1$), $j(τ)=e^{\pi \sqrt{163}}+744+196884q・・・\in \Bbb{Q}\cap \overline{\Bbb{Z}}=\Bbb{Z}$.
$744\in \Bbb{Z}$ and $196884q・・・$ is almost integer, so $j(τ)$ is almost integer.
To sum up, we used class number of $\Bbb{Q}(\sqrt{-163})$ to prove $\Bbb{Q}(j(E))=\Bbb{Q}$.
Let $K$ be an imaginary quadratic field. The proof of $[\Bbb{Q}(j(E)):\Bbb{Q}]=[K(j(E)):K]=$class number of $K$ is written in Silverman's book 'Advanced topics in the arithmetic of elliptic curves', chapter $2$, theorem $4.3$.This uses some class field theory results. $K(j(E))$ is what we call, Hilbert class field of $K$.