A short route to the main theorems of Complex Multiplication.

Class field theory classifies the abelian extension of a number (or more generally, global) field using data intrinsic to the number field – line bundles over it. This is a very powerful theory and can be used to answer a great many classical questions in algebraic number theory.

Nevertheless, the general theory offers little to no clues about generating the abelian extensions directly. In the simplest case of \mathbb Q, we can in fact explicitly parametrize the abelian extensions in terms of a simple analytic function, the exponential. In short, every abelian extension of \mathbb Q is contained in a cyclotomic field \mathbb Q(e^{2\pi i/n}) for some $n$ determined by the ramification in the abelian extension. This is a classical result predating class field theory known as the Kronecker-Weber theorem.

The problem for more general number fields looks hopelessly hard except in a few special cases. One of these is the case of quadratic imaginary number fields (that is, fields of the form \mathbb Q(\sqrt{-d}) for d > 0 and square-free. In this case, the abelian extensions are generated by torsion points on a a certain Elliptic curve. This is in analogy to the case of \mathbb Q where the abelian extensions were generated by torsion points on the one dimensional group scheme \mathbb G_m = \mathrm{Spec}\  \mathbb Q[x,x^{-1}]. This is known as complex multiplication.

Despite the parallels, the theory of complex multiplication is a lot harder than Kornecker-Weber. In fact, I believe there is no known proof of complex multiplication that does not already assume the main theorems of class field theory (in the case of quadratic imaginary number fields). For example, Silverman takes around 100 pages to develop the theory completely.

In spite of the difficult nature of the proofs, the results are very elegant and justify the effort that goes into the proofs. Hilbert has a famous quote towards this:

The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science.

I believe that there is a way to prove the main theorems of complex multiplication, undercutting much of the hard work done in Silverman’s book and I would like to explain that in this post.

The first few sections set up the general theory and follow the standard sources fairly closely. Skip directly to section 3 for the novel part.

Warning: I have not seen anyone else prove complex multiplication along these lines and it is possible that I am missing some subtlety. Proceed with caution!

Prerequisites: I will assume familiarity with the basic theory of Elliptic Curves (as in Silverman’s first book) and the main theorems of global Class Field Theory via ideals (but not their proofs!). I will gloss over many easy to prove results, especially in the early stages. A complete reference is, as mentioned above, Silverman’s second book on elliptic curves.

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Formal Summation and Dirichlet L-functions

Recall the classical Riemann zeta function:

\zeta(s) = \sum_{n\geq 1}\frac{1}{n^s}

and the Dirichlet L-functions for a character \chi: \mathbb Z \to \mathbb C:

S(s,\chi) = \sum_{n\geq 1}\frac{\chi(n)}{n^s}.

defined for \Re(s) > 1.  These functions can be analytically continued to the entire complex plane (except a pole at s=1 in the case of \zeta(s)). In particular, the values at non positive integers carry great arithmetic significance and enjoy many properties.

For instance, L(s,\chi) is an algebraic integer and in fact equal to B_{k+1,\chi}/(k+1) where B_{k,\chi} are the generalized Bernoulli numbers. Moreover, these values satisfy p-adic congruences and integrality properties such as the Kummer congruence (and generalizations to the L-functions).

The standard proof of these proceeds by showing that L(s,\chi) satisfies a functional equation that relates L(1-s,\chi) to L(s,\overline{\chi}) and then computing the values L(n,\chi) for n \geq 1 integral using analytic techniques (such as Fourier analysis).

Proving the p-adic properties is then by working directly with the definition of the generalized Bernoulli numbers instead of the L-functions. However, the L-functions are clearly the fundamental object here and it would be nice to have a way to directly work with the values at negative integers (without using the functional equation).

One might be tempted to extend the series definition to the negative integers and say:

\zeta(-k)  "="  1^k + 2^k + 3^k + \dots

For instance, consider the following (bogus) computation:

\zeta(0)\frac{t^0}{0!} = 1^0\frac{t^0}{0!} + 2^0\frac{t^0}{0!} + \dots

\zeta(-1)\frac{t^1}{1!} = 1^1\frac{t^1}{1!} + 2^0\frac{t^1}{1!} + \dots

\zeta(-2)\frac{t^2}{2!} = 1^2\frac{t^2}{2!} + 2^2\frac{t^2}{2!} + \dots

. . .

and let us “sum” the columns first:

\sum_{k\geq 0}\zeta(-k)\frac{t^{k}}{k!} = e^{t} + e^{2t} + \dots = \frac{e^t}{1 + e^t}

which, remarkably enough, is the right generating function for \zeta(-k)! We have exchanged the summation over two divergent summations and ended up with the right answer.

In fact, it is possible to rigorously justify this procedure of divergent summation and moreover, one can use it to prove a lot of arithmetic properties of these values rather easily (like the Kummer congruence). I learnt the basic method from some lecture notes of Prof. Akshay Venkatesh here: Section 3, Analytic Class Number formula and L-functions.

I then discovered that one could use these techniques to compute the explicit values (along the outline above) and prove some more stuff. I wrote this up in an article (that also explains the basic technique and should be (almost) self contained) here: Divergent Series and Dirichlet L-functions.

Noether Normalization, Spreading out and the Nullstellensatz .

Hilbert’s Nullstellensatz plays a central role in algebraic geometry. It can be seen as the fundamental link between the modern theory of schemes and the classical theory of algebraic varieties over fields. Since this is one of the first results a novice in algebraic geometry learns and is often proved very algebraically, one often does not gain a good understanding of the proof till much later.

I would like to fix my own understanding of the result and it’s geometric nature in this post. I will go through a few proofs of the theorem and point out the geometric ideas behind it. The proof of Hilbert’s lemma is usually broken up into the following two steps: 1) Prove the weak Nullstellensatz and 2) Derive the strong Nullstellensatz using the Rabinowitsch or other means. I will be focusing solely on the first step in this post. Nothing in this is new to me except perhaps the presentation and mistakes.

The weak Nullstellensatz is a statement about solving polynomial equations in multiple variables over a field. The one variable version of the problem is well understood (think Galois theory) and says that any polynomial f(x) over a field k will have all it’s solutions in some finite extension of k. The Nullstellensatz says that this result propagates to multiple variables. Continue reading

Schur’s Lemma and the Schur Orthogonality Relations.

Both Schur’s Lemma and the Schur Orthogonality relations are part of the basic foundation of representation theory. However, the connection between them is not always emphasized and the Orthogonality relations are proven more computationally.

The standard proofs of the relations never made sense to me, however there is very direct way to derive them from Schur’s Lemma (which makes perfect sense to me!) and simple facts about projections on vector spaces. More importantly, it gives a categorical interpretation of the inner product. I think this approach should be emphasized way more than it currently is and I hope this post will go a tiny way towards fixing that.

 

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The Groupoid Cardinality of Finite Semi-Simple Algebras

A groupoid is category where all the morphisms are isomorphisms and groupoid cardinality is a way to assign a notion of size to groupoids. Roughly, the idea is that one should weigh an object inversely by the number of automorphisms it has (and we only count each isomorphic object as one object).

It is important to count only one object from each isomorphism class since we want the notion of groupoid cardinality to be invariant under equivalences of groupoids (in the sense of category theory) and every category is equivalent to it’s skeleton. For further motivation for the idea of a groupoid cardinality, see Qiaochu Yuan’s post on them

This seems like quite a strange thing to do but it turns out to be quite a useful notion. One of my favorite facts about Elliptic curves is that the groupoid cardinality of the supersingular elliptic curves in characteristic p is p-1/24! See the Eichler-Deuring mass formula. 

Another interesting computation along these lines is that the number of finite sets is e. One can ask this question of various groupoids and the answer is often interesting. I will ask it today of semi-simple finite algebras of order n. By an algebra, I will always implicitly mean commutative in this post.

 

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The Weak Mordell-Weil Theorem

Let A be an abelian variety over a field K. A basic object to investigate is the group A(K). Let K be a number field. In light of the Birch and Swinnerton-Dyer conjecture and it’s relation to the class number formula, one should think of A(K) as the analog of the class group of a global field.

Thus, one might conjecture some finiteness properties of this group. It is not true that A(K) is finite as can bee seen by looking at some examples of elliptic curves but it is true that A(K) is finitely generated as an abelian group and this is the content of the Mordell-Weil Theorem.

The proof is usually broken up into two parts:

  • Weak Mordell-Weil Theorem: A(K)/nA(K) is finite for any integer n.
  • Descent using a Height Function: Deduce the full theorem from the above using a measure of size on the points of A(K).

I will focus on the first part in this section and prove it in a motivated (but sophisticated) fashion. This proof will also differ from the standard proofs in trading in for classical algebraic number theory  results (finiteness of class group and finite generation of unit group) for class field theory. This will greatly simplify the second half of the proof.

I will make free use of general theory about Abelian Varieties, Algebraic geometry and Galois Cohomology. The point of this post is not to fill in the details but to show a framework that makes the proof seem natural.

 

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Congruent Numbers and Elliptic Curves

A congruent number n is a positive integer that is the area of a right triangle with three rational number sides. In equations, we are required to find rational positive numbers a,b,c such that:

\displaystyle a^2+b^2 = c^2    and    \displaystyle n = \frac12 ab.                       (1)

The story of congruent numbers is a very old one, beginning with Diophantus. The Arabs and Fibonacci knew of the problem in the following form:

Find three rational numbers whose squares form an arithmetic progression with common difference k.

This is equivalent to finding integers X,Y,Z,T with T\neq 0 such that Y^2 - X^2 = Z^2 - Y^2 = k which reduces to finding  a right triangle with rational sides

\displaystyle \frac{Z+X}{T}, \frac{Z-X}{T}, \frac{2Y}{T}

with area k. This is the congruent number problem for k. The Arabs knew several examples of congruent numbers and Fermat stated that no square is a congruent numbers. Since we can scale triangles to assume that n is square free, this is equivalent to saying that 1 is not a congruent number.

As with many other problems in number theory, the proof of this statement had to wait four centuries for Fermat. The problem led Fermat to discover his method of infinite descent.

In more recent times, the problem has been fruitfully translated into one about Elliptic Curves. We perform a rational transformation of the defining equations (1) for a congruent number in the following way. Set x = n(a+c)/b and y = 2n^2(a+c)/b. A calculation shows that:

\displaystyle y^2 = x^3 - n^2x.                                          (2)

and y \neq 0. If y = 0, then a=-c and b = 0 but then n = \frac12 ab = 0. Conversely, given x,y satisfying (2), we find a = (x^2-y^2)/y, b = 2nx/y and c = x^2+y^2/n and one can check that these numbers satisfy (1).

The projective closure of (2) defines an elliptic curve that we will call E_n. We are interested in finding rational points on it that do not satisfy y=0. I will prove that n is a congruent number precisely when E_n has positive rank.

The proof is an interesting use of Dirichlet’s Theorem on Arithmetic Progressions and some neat ideas about Elliptic Curves and their reductions modulo primes. I will essentially assume the material in Silverman’s first book and the aforementioned Dirichlet’s Theorem.

 

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