why say complex multiplication of elliptic curves is beautiful

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.

Just as the title asked. I'm not familiar with complex multiplication. I want to know some intuitive understanding of that concept.

Any help would be appreciated.

:)

Solution 1:

Complex multiplication is a beautiful generalization of cyclotomy --- the theory of cyclotomic fields and related areas.

Cyclotomy was one of the subjects Gauss developed in Disquitiones. Some of the key points are:

• The values of the transcendental function $e^z$ at the division (i.e. torsion) points $z = \dfrac{2\pi i}{n}$ of $\mathbb C/2 \pi i \mathbb Z$ are algebraic integers (i.e. $n$th roots of unity $\zeta_n$).

• The minimal polynomial of the $\zeta_n$ is abelian (i.e. has abelian Galois group) over $\mathbb Q$.

• All abelian extensions of $\mathbb Q$ are generated by these values.

• This last fact, applied to quadratic extensions of $\mathbb Q$, can be used to derive quadratic reciprocity.

The theory of CM has a similar structure. Let me describe it here just for the field $K =\mathbb Q(i)$.

• If $\Lambda$ denotes the square lattice in $\mathbb C$, appropriately scaled, then the values of the transcendental function $\wp(z;\Lambda)$ (the Weierstrass $\wp$-function for the lattice $\Lambda$) at the points $z = \lambda/n$ for $\lambda \in \Lambda$ (which are the $n$-division, or $n$-torsion, points of $\mathbb C/\Lambda$) are algebraic integers.

• The minimal polynomials of these algebraic integers over $\mathbb Q(i)$ are abelian.

• All abelian extensions of $\mathbb Q(i)$ can be generated by these particular algebraic integers.

• This last fact can be related to reciprocity laws (e.g. biquadratic reciprocity) over $\mathbb Q(i)$.

So the theory of CM relates transcendental function theory (more particularly, the theory of elliptic functions) and algebraic number theory in an amazing way. The fact that such apparently different and disparate areas of math are connected by the theory is one of its attractions, and I'm confident that this is part (perhaps a large part) of the motivation behind Hilbert's statement.

One could note that when Langlands started writing papers explaining his expectations regarding the arithmetic of Shimura varieties --- a far reaching generalization of everything described above --- he titled one of the early expositions of his ideas Some contemporary problems with origins in the Jugendtraum (Kronecker's Jugendtraum was to develop the theory of CM). I mention this as another indication that CM theory, building on the theory of cyclotomy over $\mathbb Q$, continues to be regarded as a model for how the theories of automorphic forms, algebraic varieties, and algebraic number theory should be related together by reciprocity laws.

Solution 2:

I'll write something, probably incomplete, and hopefully someone more capable than I will fill in the gaps.

Let's start at the beginning: number fields are finite extensions of the rational field $\mathbb Q$. For every such extension $K/\mathbb Q$, we can associate a group, called the class group, which more or less measure how close the ring of integers in $K$ is to being a Unique Factorization Domain. In particular, when we talk of abelian extensions of $\mathbb Q$, we mean that the class group the Galois group $\mathrm{Gal}(K/\mathbb Q)$ is an abelian group.

Now, it turns out that the abelian extensions of $\mathbb Q$ have a very nice characterization, through the Kronecker-Weber theorem:

All abelian extensions of $\mathbb Q$ are subfields of a cyclotomic field $\mathbb Q(\zeta_n)/\mathbb Q$.

The problem is that, this only holds for abelian extensions of the rationals. The next question number theorists in the 19th century tried to answer is:

How can we extend this to the next simplest case, that of quadratic fields?

Recall that all quadratic extensions of $\mathbb Q$ are of the form $\mathbb Q(\sqrt{d})$, where $d$ is a square-free integer. If it is positive we get a real quadratic field; if it is negative, an imaginary quadratic field. The theory of complex multiplication is therefore a way to generalize the Kronecker-Weber theorem to imaginary quadratic fields. But you may ask, where do the elliptic curves come in? Through the following theorem:

Let $E$ be an elliptic curve over $\mathbb C$. Then the ring of endomorphisms of $E$ is either $\mathbb Z$ or isomorphic to an order in an imaginary quadratic field.

Therefore, elliptic curves with CM (i.e. whose ring of endomorphisms is larger than $\mathbb Z$) are connected to imaginary quadratic fields. Moreover, for any such field and any order $\mathcal O$ in it, there exists an elliptic curve $E$ such that $End(E)\cong\mathcal O$. And this is, somehow, the first step in generating all abelian extensions of imaginary quadratic fields (bit to go further would need class field theory).

In conclusion: the reason why Hilbert was so fond of the theory of Complex Multiplication is probably that it gives a rather elegant solution to the problem of "extending" the Kronecker-Weber theorem to imaginary quadratic fields.