Iwasawa Theory and the analogy with Function Fields

Fix K a number field and p a prime. Define K_n = K(\mu_{p^n}) where \mu_k are the k-th roots of unity and let K_\infty = \bigcup_{n\geq 1}K_n.  Let C_n be the class group of K_n and let A_n be the p-part of C_n. Finally, let G_n be the Galois group of K_n/K.

Iwasawa theory broadly describes the structure of A_n  and the action of G_n on it. For instance, Iwasawa proved in 1959 that the size of A_n is equal to p^{\lambda n+p^n\mu+\nu} for large n.

There are also various results relating A_n to the p-adic properties of special values of the zeta function of K. For instance, taking K = \mathbb Q, let \zeta(s) be the usual zeta function. For negative odd integers n, \zeta(n) is always a rational number and writing these in reduced form, Kummer proved that p divides one of the numerators of \zeta(-2n-1) if and only if A_1 is non trivial. The Herbrand-Ribet theorem provides a generalization of this statement and also describes the action of the Galois group on A_n.

While very beautiful and striking, these results might seem to come out of nowhere. I would like to describe Iwasawa’s original motivation in the reminder of this post.

The analogy with Function Fields:

Consider a function field over a finite field or equivalently, a projective curve C over a finite \mathbb F_q. There are strong similarities between these and number fields and the correct analog of the Ideal Class group is the group of divisor classes of degree 0. This group can be given the structure of an Abelian variety and is called the Jacobian of C and is denoted J(C). It has dimension equal to the genus of the curve g.

We understand this group well. In particular, if we take p to be a prime different from the characteristic of the base field, then the p-torsion of J(C) (over an algebraic closure \overline{\mathbb F_q} ) is isomorphic to (\mathbb Q_p/\mathbb Z_p)^{2g}.

However, we are interested in the \mathbb F_q points of J(C). Luckily, the image of the natural embedding J(C)(\mathbb F_q) \to J(C)(\overline{\mathbb F_q}) is contained in the torsion subgroup which is well understood.

Since \overline{\mathbb F_q} is obtained from \mathbb F_q by adjoining all the roots of unity, one might hope that a similar procedure might help us on the number theoretic side. Unfortunately, such a field is too large and loses too much arithmetic information.

However, since the ideal class group C(K) is a finite abelian group for a number field, it is determined by it’s p-Sylow subgroups. Fixing a prime p as before, let A(K) denote the p-part of C(K). It turns out that we can study A(K) by adjoining all the p-th roots of unity.

That is, in the notation from the beginning of the post, if we define A(K_\infty) = \varinjlim A_n as a limit of directed sets, then it can be shown that

\displaystyle A(K_\infty) \cong (\mathbb Q_p/\mathbb Z_p)^\lambda\oplus((\oplus^\infty A)/B).

Here, \lambda is a non-negative integer, A is a finite abelian group of order some power of p, the direct sum is of an infinite copies of A and B is a finite subgroup of \oplus^\infty A.

It is conjectured that in general, A does not appear and we can prove this in the case that K/\mathbb Q is an abelian extension.

The Frobenius and the action of the Galois Group:

In the case of function fields, we have a Frobenius action on C and on J(C) whose fixed points are precisely the \mathbb F_q points. The action of the Frobenius on J(C)(\overline{\mathbb F_q}) \cong (\mathbb Q_p/\mathbb Z_p)^{2g} induces an endomorphism on it which may be represented by a matrix A_C \in M_{2g}(\mathbb Z_p). Importantly, the zeta function of C over \mathbb F_q can be expressed in terms of the characteristic polynomial of A_C. This is the content of the Weil Conjectures.

Let us come back to number fields and K = \mathbb Q for simplicity. As mentioned in the last section, A_\infty \cong (\mathbb Q_p/\mathbb Z_p)^\lambda. The Galois group G_\infty acts on $A_\infty$ naturally but unlike the function field case, the Galois group does not have a generator. Since G_n \cong \mathbb (Z/p^n\mathbb Z)^\times, we have G_\infty \cong \mathbb Z_p^\times.

We have \mathbb Z_p^\times \cong (\mathbb Z/p\mathbb Z)^\times\times (1+p\mathbb Z_p). The second factor is isomorphic to \mathbb Z_p and we take a topological generator u of it.

If we believe the analogy with function fields, then we would expect a relation between the characteristic polynomial of u and some zeta function. The answer is, amazingly enough, yes! This zeta function is called the Kubota-Leopolft p-adic L function derived from the special values of the ordinary L-functions at integers.

The Iwasawa Main Conjecture:

To make everything above more precise, define the Teichmuller character:

\omega: \mathbb Z/p\mathbb Z \to \mathbb Z_p = G_\infty

by the canonical splitting (\mathbb Z_p)^\times = (\mathbb Z/p\mathbb Z)^\times\times (1+p\mathbb Z_p). Extend \omega to be a map on G_\infty through the quotient \mathbb Z_p^\times \to (\mathbb Z/p\mathbb Z)^\times. For an integer i in the range 0\leq i<p-1, define:

A^i_\infty = \{x \in A_\infty : \sigma(x) = \omega(\sigma)^ix \text{ for all } \sigma \in \Delta\}. By general theory, we have:

 \displaystyle A_\infty = \bigoplus_{i=0}^{p-2}A_\infty^i.

Write A_\infty^i \cong (\mathbb Q_p/\mathbb Z_p)^{\lambda_i}. Our topological generator u acts on A_\infty^i and the action of u-1 can be represented by a matrix A_i of degree \lambda_i. Let \varphi_i(T) be the characteristic polynomial of A_i.

The Iwasawa main conjecture states the relation between the polynomial \varphi_i(T) defined as above and the p-adic Riemann \zeta function:

Theorem: Let i be an odd integer in the range 1<i<p-1.

  1. For any negative integer m satisfying m\equiv i \pmod p-1 , there exists a unique formal power series g_i(T) \in \mathbb Z_p[[t]] such that g_i(u^m-1) = (1-p^{-m})\zeta(m) where \zeta(s) is the Riemann \zeta function and $latex $ u is our topological generator as before. The function s \to g_i(u^s-1) is a p-adic continuous function from \mathbb Z_p to \mathbb Z_p and is the p-adic L-function we alluded to before.
  2. (Iwasawa Main Conjecture, Theorem of Mazur-Wiles) The characteristic polynomial \varphi_i(t) defined as above and g_i(t) coincides up to unit in \mathbb Z_p[[t]]

This was proven in 1984 by Mazur and Wiles for \mathbb Q and for totally real number fields in 1990 by Wiles.

 

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.


So far, we know that n is a congruent number if and only if E_n has rational points (x,y) with y \neq 0. Recall that for an elliptic curve in the standard Weierstrass form (as in (2)), y = 0 if and only if (x,y) has 2-torsion.

Therefore, our problem reduces to showing that the only torsion of E_n is 2-torsion for all n. In fact, we always have non-trivial $2$-torsion and the points are given by (0,0), (n,0),(-n,0) and the point at infinity.

Denote the m-torsion my E_n[m]. The rough outline of the proof is as follows:

  1. E_n[m] maps injectively into the reduction of E_n modulo a prime p for all but finitely many primes.
  2. The number of \mathbb F_p points of E_n is independent of p and equal to p+1 whenever p \equiv 3 \pmod 4.
  3. This would imply that m|p+1 for a set of primes $p$ of density $1/2$ but by Dirichlet’s Unit Theorem, the set of such primes is of density 1/\varphi(m) < 1/2 for all m>4.

Step 1:

Recall that the size of E_n[m](\mathbb Q) is finite (and has exactly m^2 elements in \overline{\mathbb Q}. Also, recall that if (x_1,y_1,z_1) and (x_2,y_2,z_2) are two points in the projective plane (over any field k), then they are equal if and only if

\displaystyle x_1y_2-x_2y_1, x_1z_2-x_2z_1, y_1z_2-y_2z_1    (3)

are all 0. Thinking of our E_n has embedded in the projective plane, reduction modulo p is simply reduction on each of the co-ordinates. Therefore, if we pick any finite set of \mathbb Q points on $E_n$, then for any prime p greater than any of the prime divisors of (3) , it’s reduction will be non-zero.

Since E_n[m] is a group, this is sufficient to show that the reduction map is injective for all but finitely many primes p (once we fix m).

Step 2:

Fix a prime p. One could prove this by an explicit calculation involving quadratic characters but there is a neater way assuming some knowledge about the endomorphism ring R_n =   \mathrm{End}_{\mathbb F_p}(E_n). The relevant facts are the following:

  1. Over a finite field \mathbb F_q, the endomorphism ring is either an order in a quadratic imaginary field or an order in a quaternion algebra.
  2. Further, over \mathbb F_p with p>5, the latter case occurs precisely when the number of elements on the curve is p+1 and the curve is called supersingular.
  3. In either case, for any endomorphism f, there is a dual \hat f such that N(f) =  f\circ\hat f is multiplication by the degree of f, which is always non-negative.

A few words about the above statements: They are in true for all Elliptic curves and not just E_n. A reference for all of the above is the chapter on Finite Fields in Silverman’s first book on Elliptic Curves.

(3) is true over any field. The function N occurring in (3) is the usual norm function on a number field or a quaternion algebra.

One can show using the Tate Module that the dim of R_n\otimes \mathbb Q is at most 4 over \mathbb Q. Since there is always a Frobenius element \varphi such that N(\varphi) = p, this rules out the case of \mathbb Z where the N function maps to the squares. This proves (1).

A sufficient (and necessary) condition for R_n being 4 dimensional is that \hat\varphi be inseparable but this implies a_p = \varphi + \hat\varphi \equiv 0 \pmod p. As a consequence of the Hasse-Weil inequality which says a_p \leq 2\sqrt p, this would show that a_p = 0 p>5. One further proves that |E_n(\mathbb F_p)| = p+1-a_p which shows (2).

Now, given the (3) statements, it is easy to complete step 2. Note that (x,y,z) \to (-x,iy,z) is always an endomorphism of E_n as long as i = \sqrt{-1} is defined in \mathbb F_p, equivalently, as long as p\equiv -1 (4). Therefore, if R_n were two dimensional, it would have to be an order in \mathbb Z[i].

As mentioned before, we also have the Frobenius \mathbb \varphi with norm p. This shows that \varphi is a prime in the ring. However, the norm of any prime element in \mathbb Z[i] is either a square or congruent to 1 modulo 4. This forces R_n to be 4-dimensional and we are done by (2).

Step 3:

If E_n(\mathbb Q) has any torsion point P that is not 2-torsion, then we can find some point Q of order more than 4. Let this order be m.

By step (1), Q has order m for all but finitely many primes and therefore, since E_n(\mathbb F_p) is a group of size p+1 (for p\equiv 4\pmod 4 by step (2)), m|p+1 for all but finitely many primes p \equiv -1 \pmod 4.

Since the (Dirichlet) density of primes of the form 3k+4 has density 1/2 and a finite set of primes does not contribute to (Dirichlet) density, we can put this differently in the following way:

The  Dirichlet density of primes p \equiv -1\pmod m is at least 1/2.

However, we know that the density of primes of the form km+1 is exactly 1/\varphi(m) which is strictly less than 1/2 whenever m > 4. This provides the required contradiction and completes the proof.

The statements about the densities of various primes above all follow from Dirichlet’s Theorem on Arithmetic Progressions.


 

I would like to end by talking about a conjectural algorithm to detect congruent numbers. It is called Tunnell’s algorithm and is based on the idea above: A number n is congruent if and only if E_n(\mathbb Q) has positive rank.

The algorithm is easy to describe (and execute) but it’s correctness depends on a very deep theorem about Elliptic Curves (the Birch and Swinnerton-Dyer conjecture).

The algorithm is as follows: For a square free integer n, define:

{\begin{matrix}A_{n}&=&\#\{(x,y,z)\in {\mathbb  {Z}}^{3}|n=2x^{2}+y^{2}+32z^{2}\}\\B_{n}&=&\#\{(x,y,z)\in {\mathbb  {Z}}^{3}|n=2x^{2}+y^{2}+8z^{2}\}\quad \\C_{n}&=&\#\{(x,y,z)\in {\mathbb  {Z}}^{3}|n=8x^{2}+2y^{2}+64z^{2}\}\\D_{n}&=&\#\{(x,y,z)\in {\mathbb  {Z}}^{3}|n=8x^{2}+2y^{2}+16z^{2}\}.\end{matrix}}

Tunnell proved unconditionally that if n is an odd congruent number, then 2A_n = B_n and if n is an even congruent number, then 2C_n = D_n. The converse is true assuming the Birch Swinnerton-Dyer conjecture.

More precisely, if we denote by L (s)= L(E_n,s) the L-function for E_n over $\mathbb Q$ at s=1, recall that the Birch-Swinnerton Dyer conjecture states that the order of vanishing of L(s) at s = 1 is the rank of E_n(\mathbb Q).

Therefore, n is a congruent number if and only if L(E_n,s) = 0. What Tunnel showed was that:

L(E_n) = \begin{cases}\gamma(2A_n-B_n) & n \text{is odd}\\\gamma(2C_n-D_n) & n \text{ is even}\end{cases}

where \gamma is a non-zero constant. Note that E_n always has complex multiplication over \mathbb Q since (x,y,z) \to (-x,iy,z) is an automorphism of order 4.

The unconditional direction of Tunnel’s criterion follows from the following theorem of Coates-Wiles in 1976:

Theorem[Coates-Wiles(1976)] If an Elliptic Curve over \mathbb Q has complex multiplication by a ring of integers with class number 1 and has positive rank over \mathbb Q, then the corresponding L-function vanishes at s=1.

 

Descent on Vector Spaces and Cohomology

It is quite often of interest to study the properties of some variety {X/\mathbb Q}. However, it is generally much easier to study varieties over algebraically closed fields and so we need some way of translating a property of {X_{\overline{\mathbb Q}}/\overline{\mathbb Q}} to {X/\mathbb Q} . This idea is known as descent and in this post, I would like to say a little bit about the simplest example of descent – over vector spaces.

Let {L/K} be an extension of fields and {V} a vector space over {K} . Consider {W = V\otimes_K L } . The Galois Group {G = \mathop{Gal}(L/K)} acts on {W} through the second factor one can consider the {K} -vector space {W^G} . This is the vector space fixed by {G} .

Theorem 1 (Descent of Vector Spaces) The natural map {W^G\otimes_K L \rightarrow W} is an isomorphism. In particular, if {W} is finite dimensional, then {\dim_K W^G = \dim_L W} .

It is not hard to prove this theorem directly but I would like to relate it to another theorem. This is also well known and is a generalization of the famous Hilbert’s Theorem 90. Let {{GL}_n(L)} be the group of invertible {n} -dimensional matrices over {L} and consider the cohomology {H^1(G,{GL}_n(L))} . This is not a group unless {n=1} since {{GL}_n(L)} is non-commutative in general. However, it is a pointed set and we have the following theorem:

Theorem 2 (Hilbert’s 90) \displaystyle H^1(G,{GL}_n(L)) = \{0\}.

Hilbert stated the above theorem (in a disguised form) for {n=1} and {L/K} a finite cyclic extension. Noether generalized the theorem to arbitrary extensions. I do not know who is responsible for the generalization to general linear groups but I saw this theorem first in Serre’s “Galois Cohomology”.

In this post, I will show that the above theorems are equivalent on the following sense:

Let {W} be a {L} -vector space. We will say that a group {G} acts semi-linearly on it if \sigma(lv) = \sigma(l)\sigma(v) \text{ for all }\sigma \in G.

The typical example is when {G = \mathop{Gal}(L/K)} acts co-ordinate wise on {W = L^n} or equivalently {W = V\otimes_K L} for a {K} -vector space {V} . We will show that this is essentially the only example by proving:

Theorem 3 There is a bijection:

\displaystyle H^1(G,{GL}_n(L)) \longleftrightarrow \frac{\{\text{n-dimensional L- vector spaces with semilinear G-action}\}}{\text{isomorphisms}}

Proof: I will use {x^g} to denote {g} acting on {x} throughout:

Let us first establish the maps. Given a 1-cocyle {\eta:G \rightarrow \ mathop{GL}_n(L)} , let the corresponding vector space {W_\eta} be {L^n} with the action for {(g,w) \in G\times W} being given by {(g,w) \rightarrow \eta_g(w^g)} where {w^g} stands for the action of {g} co-ordinate wise. It is easy to verify that this is well defined:

Given {l\in L} , {\eta_g((lw)^g) = \eta_g(l^gw^g) = l^g\eta_g(w^g)} . This shows that the action is semi-linear. Then, given a one-cocycle cohomologous to {\eta} (that is, given {\tau_g = A^{-1}\eta_g A^g} ), we have the following isomorphism of vector spaces with group actions given by:

\displaystyle W_\eta \rightarrow W_\tau, w \rightarrow A^{-1}w.

That is, {A\tau_g((A^{-1}w)^g) = \eta_g(w^g)} as can be easily checked. This establishes that {\eta \rightarrow W_\eta} is well defined.

To construct the inverse, let {W} be a {L} -vector space with a semilinear {G} -action. Fix a basis {e_1,\dots, e_n} . Denote the column vector corresponding to this basis as {[e]} . Define {\eta_g} to be the unique transformation such that {\eta_g[e]^g = [e]} . To check that this is a 1-cocycle, note that:

\displaystyle \eta_g\eta_h^g[e]^{gh} = \eta_g(\eta_h[e]^h)^g = \eta_g[e]^g = [e]

and hence by uniqueness {\eta_{gh} = \eta_g\eta_h^g} .

Finally, it is easy to verify that these maps really are inverses. \Box

To see that we really have shown that Theorem 1 and Theorem 2 are equivalent, note that Theorem 1 is clearly true for {L^n = K^n\otimes_KL} . Then, if {H^1(G,{GL}_n(L)) = \{0\}} , there is a unique {L} -dimensional vector space with a semi-linear action and we can check Theorem {1} on this unique vector space.

Conversely, if Theorem {1} is true, then {H^1(G,{GL}_n(L))} is a one-element set.

Proofs for Theorem 1 and Theorem 2 can be found in many places. Serre’s Galois Cohomology is a good place to read about Group Cohomology generally and Theorem 2 in particular.

UPDATE: I later discovered that the content of this post is Exercise 1.9 in Poonen’s “Rational Points on Varieties”. He also proves Theorem 1 as Lemma 1.3.10.