# Elliptic functions and “complex multiplication” [closed]

I have before me a copy of "The Indian Mathematician Ramanujan", by G. H. Hardy (not actually related to me, as far as I know), which appeared in volume 44, number 3 (March 1937) of The American Mathematical Monthly, on pages 137‒155.

One of the theorems of Ramanujan stated there is this:

If $\displaystyle F(k) = 1 + \left( \frac 1 2 \right)^2 k + \left( \frac{1\cdot3}{2\cdot4} \right)^2 k^2 + \cdots$ and $F(1-k) = \sqrt{210} F(k)$, then \begin{align} k = {} & (\sqrt 2 - 1 )^4 (2-\sqrt 3)^2(\sqrt7 - \sqrt 6)^4 (8-3\sqrt7)^2 \\ & \cdot (\sqrt{10} - 3)^4(4-\sqrt{15})^4(\sqrt{15}-\sqrt{14})^2 (6-\sqrt{35})^2. \tag{13} \end{align}

Of this Hardy wrote:

An expert on elliptic functions can see at once that $(13)$ is derived somehow from the theory of "complex multiplication", [ . . . ]

What does the phrase "complex multiplication" mean in this context?

## closed as off-topic by Steven Landsburg, Qiaochu Yuan, Stefan Kohl, Alexey Ustinov, Chris GodsilOct 27 '16 at 12:32

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Steven Landsburg, Qiaochu Yuan, Stefan Kohl, Alexey Ustinov, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

• In 1999 at M.I.T. I met Robert Kanigel, who, noting my last name and my affiliation with the math department, asked if I was related to "G.H."? I said not as far as I know. On the following day I learned that he was Ramanujan's biographer, author of The Man Who Knew Infinity. Far more recently a movie inspired by that book appeared, with Jeremy Irons and Dev Patel. – Michael Hardy Oct 27 '16 at 5:33
• Complex Multiplication in the context of Elliptic Curves refers to Elliptic curves that have "extra" endomorphisms. Usually, Elliptic curves over a characteristic 0 field (C for example), have endomorphism ring $\Bbb Z$. However sometimes it happens that you have more endomorphisms. An example is $y^2 = x^3-x$ and has ring Z[i]. – Asvin Oct 27 '16 at 6:13
• This is not research-level, and in particular could have been resolved by googling 'complex multiplication' and clicking on the first result. – Kevin Casto Oct 27 '16 at 6:35
• This question is OK; a specific equation is derived from complex multiplication, so the issue is: how? Which imaginary quadratic extension of $\mathbb{Q}$ is involved? – Venkataramana Oct 27 '16 at 6:49