Kolyvagin's Cohomology Classes

In this section we define Kolyvagin's cohomology classes. Later we will explain the properties that these classes have, and eventually use them to sketch a proof of finiteness of Shafarevich-Tate groups of certain elliptic curves.

We will use, when possible, similar notation to the notation Kolyvagin uses in his papers (e.g., [Kol91]). If $A$ is an abelian group let $A/M = A/(MA)$. Kolyvagin writes $A_M$ for the $M$-torsion subgroup, but we will instead write $A[M]$ for this group.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with no constraint on the rank of $E$. Fix a modular parametrization $\pi:X_0(N)\to E$, where $N$ is the conductor of $E$. Let $K$ be a quadratic imaginary field with discriminant $D$ that satisfies the Heegner hypothesis for $E$, so each prime dividing $N$ splits in $K$, and assume for simplicity that $D\neq -3, -4$.

Let $\O_K$ be the ring of integer of $K$. Since $K$ satisfies the Heegner hypothesis, there is an ideal $\mathcal{N}$ in $\O_K$ such that $\O_K/\mathcal{N}$ is cyclic of order $N$. For any positive integer $\lambda$, let $K_{\lambda}$ be the ray class field of $K$ associated to the conductor $\lambda$ (see Definition 3.13). Recall that $K_{\lambda}$ is an abelian extension of $K$ that is unramified outside $\lambda$, whose existence is guaranteed by class field theory. Let $\O_{\lambda} = \mathbb{Z}+ \lambda \O_K$ be the order in $\O_K$ of conductor $\lambda$, and let $\mathcal{N}_{\lambda} = \mathcal{N}\cap \O_{\lambda}$. Let

$$z_{\lambda} = [(\mathbb{C}/ \O_{\lambda}, \mathcal{N}_{\lambda}^{-1} / \O_{\lambda})] = X_0(N)(K_{\lambda})$$

be the Heegner point associated to $\lambda$. Also, let

$$y_{\lambda} = \pi(z_\lambda) \in E(K_{\lambda})$$

be the image of the Heegner point on the curve $E$.

Let $R=\End (E/\mathbb{C})$, and let $B(E)$ be the set of primes $\ell \geq 3$ in $\mathbb{Z}$ that do not divide the discriminant of $R$ and are such that the image of the representation

$$\rho_{E,\ell} : \Gal (\overline{\mathbb{Q}}/\mathbb{Q}) \to \Aut (\Tate _{\ell}(E))$$

contains $\Aut _R(\Tate _{\ell}(E))$, where $\Aut _R(\Tate _{\ell}(E))$ is the set of automorphisms that commute with the action of $R$ on $\Tate _{\ell}(E)$. Note that if $\ell \geq 5$ the condition that $\rho_{E,\ell}$ is surjective is equivalent to the simpler condition that

$$\overline{\rho}_{E,\ell} : \Gal (\overline{\mathbb{Q}}/\mathbb{Q}) \to \Aut _{R}(E[\ell])$$

is surjective. The set $B(E)$ contains all but finitely many primes, by theorems of Serre [Ser72], Mazur [Maz78], and CM theory, and one can compute $B(E)$.

sage: E = EllipticCurve('11a')
sage: E.non_surjective()
[(5, '5-torsion')]
sage: E = EllipticCurve('389a')
sage: E.non_surjective()
[]

Fix a prime $\ell\in B(E)$. We next introduce some very useful notation. Let $\Lambda^1$ denote the set of all primes $p\in\mathbb{Z}$ such that $p\nmid N$, $p$ remains prime in $\O_K$, and for which

$$n(p) = \ord _{\ell}(\gcd(p+1, a_p)) \geq 1.$$

For any positive integer $r$, let $\Lambda^r$ denote the set of all products of $r$ distinct primes in $\Lambda^1$; by definition $\Lambda^0 = \{ 1\}$. Finally, let

$$\Lambda = \bigcup_{r\geq 0} \Lambda^r.$$

For any $r > 0$ and $\lambda \in \Lambda^r$, let

$$n(\lambda) = \min_{p\mid \lambda} n(p)$$

be the ``worst'' of all the powers of $p$ that divide $\gcd(p+1,a_p)$. If $\lambda = 1$, set $n(\lambda)=+\infty$.

Fix an element $\lambda \in \Lambda$, with $\lambda\neq 1$, and consider the $\ell$-power

$$M = M_{\lambda} = \ell^{n(\lambda)}.$$

Recall from Section 2.2.1 that we associate to the short exact sequence

$$0 \to E[M] \to E \xrightarrow{[M]} E \to 0$$

an exact sequence

$$0 \to E(K)/M \to \H^1(K,E[M]) \to H^1(K,E)[M]\to 0.$$

Our immediate goal is to construct an interesting cohomology class

$$c_{\lambda} \in \H^1(K,E[M]).$$

If $L/K$ is any Galois extension, we have (see Section 2.1.2 for most of this) an exact sequence

$$0 \to \H^1(L/K, E[M](L)) \to \H^1(K, E[M]) \to \H^1(L,E[M])^{\Gal (L/K)} \to 0.$$ (3.4.1)

Lemma 3.23   We have $E[M](K_{\lambda}) = 0$.

Thus (3.4.1) with $L=K_{\lambda}$ becomes

$$\H^1(K, E[M]) \xrightarrow{\quad\cong \quad} \H^1(K_{\lambda},E[M])^{G_{\lambda}}$$ (3.4.2)

where $G_{\lambda} = \Gal (K_{\lambda}/K)$. Putting this together, we obtain the following commutative diagram with exact rows and columns:

$$\xymatrix{ 0 \ar[r]& {(E(K_{\lambda})/M)^{G_{\lambda}}} \ar... ... 0\\ & & & \H ^1(K_\lambda/K, E)[M]\ar@{^(->}[u]^{\inf}\\ }$$

Thus to construct $c_\lambda \in \H^1(K,E[M])$, it suffices to construct a class $c_{\lambda}' \in \H^1(K_{\lambda},E[M])$ that is invariant under the action of $G_{\lambda}$. We will do this by constructing an element of $E(K_{\lambda})$ and using the inclusion

$$E(K_{\lambda})/M \hookrightarrow \H^1(K_{\lambda}, E[M]).$$ (3.4.3)

In particular, we will construct an element of the group $E(K_{\lambda})/M$ that is invariant under the action of $G_{\lambda}$.

Recall that $y_\lambda \in E(K_{\lambda})$. Unfortunately, there is no reason that the class

$$[y_\lambda]\in E(K_{\lambda})/M$$

should be invariant under the action of $G_{\lambda}$. To deal with this problem, Kolyvagin introduced a new and original idea which we now explain.

Let $H=K_1$ be the Hilbert class field of $K$. Write $\lambda = p_1 \cdots p_r$, and for each $p = p_i$ let $G_{p} = \Gal (K_{p}/K)$ where $K_{p}$ is the ray class field associated to $p$. Class field theory implies that the natural map

$$\Gal (K_\lambda/K_1) \cong \to G_{p_1} \times G_{p_2} \times \cdots \times G_{p_r}$$

is an isomorphism. Moreover, each group $G_{p_i}$ is cyclic of order $p_i + 1$. For each $p = p_i$, let $\sigma_p$ be a fixed choice of generator of $G_{p}$, and let

$$\Tr _{p} = \sum_{\sigma \in G_{p}} \sigma \in \mathbb{Z}[G_{p}].$$

Finally, let $D_{p} \in \mathbb{Z}[G_p]$ be any solution of the equation

$$(\sigma_p - 1) \cdot D_p = p + 1 - \Tr _p.$$ (3.4.4)

For example, Kolyvagin always takes

$$D_p = \sum_{i=1}^p i \sigma_p^i = -\sum_{i=1}^{p+1} (\sigma_p^i - 1)/(\sigma_p-1).$$

Notice that the choice of $D_p$ is well defined up to addition of elements in $\mathbb{Z}\Tr _p$. Let

$$D_\lambda = \prod D_p = D_{p_1} \cdot D_{p_2} \cdot \cdots \cdot D_{p_r} \in \mathbb{Z}[G_\lambda].$$

Finally, let $S$ be a set of coset representatives for $\Gal (K_\lambda/K_1)$ in $G_{\lambda} = \Gal (K_{\lambda}/K)$, and let

$$J_{\lambda} = \sum_{\sigma \in S} \sigma \in \mathbb{Z}[G_{\lambda}].$$

Let

$$P_\lambda = J_{\lambda} D_\lambda y_\lambda \in E(K_{\lambda}).$$

Note that if $\lambda = 1$, then $K_{\lambda} = K_1$, so

$$P_1 = J_1 y_{\lambda} = \Tr _{K_1/K}(y_\lambda) = y_K \in E(K).$$

Before proving that we can use $P_{\lambda}$ to define a cohomology class in $\H^1(K,E[M])$, we state two crucial facts about the structure of the Heegner points $y_\lambda$.

Proposition 3.24   Write $\lambda = p \lambda'$, and let $a_p = a_p(E) = p+1 - \char93 E(\mathbb{F}_p)$.

1. We have
   
   $$\Tr _p(y_\lambda) = a_p y_{\lambda'}$$
   
   
   in $E(K_{\lambda'})$.
2. Each prime factor $\wp_{\lambda}$ of $p$ in $K_{\lambda}$ divides a unique prime $\wp_{\lambda'}$ of $K_{\lambda'}$, and we have a congruence
   
   $$y_{\lambda} \equiv \Frob (\wp_{\lambda'})(y_{\lambda'}) \pmod{\wp_{\lambda}}.$$
   
   

Proposition 3.25   The class $[P_\lambda]$ of $P_\lambda$ in $E(K_{\lambda})/M$ is fixed by $G_\lambda$.

We have now constructed an element of $E(K_\lambda)/M$ that is fixed by $G_{\lambda}$. Via (3.4.3) this defines an element $c'_{\lambda} \in \H^1(K_{\lambda}, E[M])$. But then using (3.4.2) we obtain our sought after class $c_{\lambda} \in \H^1(K, E[M])$.

We will also be interested in the image $d_{\lambda}$ of $c_{\lambda}$ in $\H^1(K,E)[M]$.

Proposition 3.26   If $v$ is archimedean or $v\nmid \lambda$, then

$$\res _v(d_{\lambda}) = 0.$$

Proposition 3.27   Write $\lambda = p m$ and let $\wp = p\O_K$ be the unique prime ideal of $K$ dividing $p$. Let $v$ be a place of $K_m$ that divides $\wp$. Then the order of $\res _\wp(d_\lambda)$ is the same as the order of

$$[P_m]\in E(K_\wp)/M E(K_{\wp}),$$

where $K_\wp$ denotes the completion of $K$ at $\wp$. (Note that $\wp$ splits completely in $K_m/K$ by class field theory, since $\wp = p\O_K$ is principal and coprime to $m$, so $P_m \in E(K_\wp)$.)

Next we consider a consequence of Proposition 3.27 when $y_K$ is not a torsion point. Note that $y_K$ nontorsion implies that $y_K \not \in M E(K)$ for all but finitely many $M$. Moreover, the Gross-Zagier theorem implies that $y_K$ is nontorsion if and only if $\ord _{s=1} L(E,s) \leq 1$.

Proposition 3.28   Suppose that $y_K \in E(K)$ is not divisible by $M$. Then there are infinitely many $p\in\Lambda^1$ such that $d_{p} \in \H^1(K,E)[M]$ is nonzero.

Remark 3.29   See, e.g., [Ste02] for an application of this idea to a problem raised by Lang and Tate in [LT58].

Theorem 3.30 (Kolyvagin)   Suppose $E$ is a modular elliptic curve over $\mathbb{Q}$ and $K$ is a quadratic imaginary field that satisfies the Heegner hypothesis for $E$ and is such that $y_K \in E(K)$ is nontorsion. Then $E(K)$ has rank $1$ and

$$\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontse... ...{n}\selectfont Sh}}}(E/K) \mid b \cdot [E(K):\mathbb{Z}y_K]^2,$$

where $b$ is a positive integer divisible only by primes $\ell\in B(E)$ (i.e., for which the $\ell$-adic representation is not as surjective as possible).

After Kolyvagin proved his theorem, independently Murty-Murty, Bump-Friedberg-Hoffstein, Waldspurger, each proved that infinitely many such quadratic imaginary $K$ always exists so long as $E$ has analytic rank $0$ or $1$. Also, Taylor and Wiles proved that every $E$ over $\mathbb{Q}$ is modular. Thus we have the following theorem:

Theorem 3.31   Suppose $E$ is an elliptic curve over $\mathbb{Q}$ with

$$r_{E,{\rm an}} = \ord _{s=1}L(E,s) \leq 1.$$

Then $E(\mathbb{Q})$ has rank $r_{E,{\rm an}}$, the group ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is finite, and there is an explicit computable upper bound on $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$.

The author has computed the upper bound of the theorem for all elliptic curves with conductor up to $1000$ and $r_{E,{\rm an}}\leq 1$.