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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:

  1. Why is the $j$-function modular? What's the difference between a modular form and a modular function?
  2. Why is the class number equal to the number of distinct similar quadratic forms?
Zongshu Wu
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    look through these https://math.stackexchange.com/search?q=163 – Will Jagy Apr 16 '23 at 02:43
  • The post you linked is helpful but it doesn't answer why the j-function is modular and WHY j((b+√-D)/2a) behaves like it does. Moreover, what do quadratic forms have to do with the class number? – Zongshu Wu Apr 16 '23 at 03:24
  • You're trying to ask several different questions in this post: (1) What is the difference between a modular function and a modular form? (2) What is the intuition behind the definition of the $j$-invariant? (3) What is the connection between the $j$-invariant and CM theory? These are separate questions and should be made into separate posts. (They also have little to do with the title of your question.) – Viktor Vaughn Apr 16 '23 at 03:34
  • ok yeah I was so confused I couldn't even ask the question properly. nvm this dumpster fire – Zongshu Wu Apr 16 '23 at 03:35
  • @JyrkiLahtonen . The questioner ln your (fine) reference calls it Ramanujan's constant. But a 19th-century mathematician, who performed as a calculating prodigy as a boy, observed very late in life that $\exp (\pi\sqrt {163})$ differs from an integer by less than $10^{-12}.$ Some of my books are in storage, but my reference is Rousse-Ball & Coxeter, Mathematical Recreations And Essays, 11th edition, in the chapter "Three Calculating Prodigies". – DanielWainfleet Apr 16 '23 at 03:58

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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 $K$.

Pont
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  • Oops. Can't understand what is going on. I am a high schooler with only knowledge of some basic complex analysis and algebraic number theory and no idea what class field theory or Complex Multiplication is all about. – Zongshu Wu Apr 16 '23 at 03:29
  • Sorry, $q=e^{2\pi τi}$. Detailed thing is written in Silverman's book, 'elliptic curve theory over $\Bbb{C}$' ＋ 'Class field theory' gives you the result. Imaginary quadratic field is closely related to $CM$ elliptic curves. – Pont Apr 16 '23 at 03:37