This short article gives a surprisingly complete account of Kolyvagin's proof that if E is an elliptic curve with analytic rank 0 whose L-function has an appropriate nonvanishing twist (this is always true), then the Mordell-Weil and Shafarevich-Tate groups of E are finite.

Here is a 3MB scan of the article in PDF format.

Let $N$ be a positive integer and let $X_0(N)$ denote the modular curve
associated to the group $\Gamma_0(N)=\{\left(\smallmatrix a&b\\ c&d\endsmallmatrix\right)\in\roman{SL}_2(\bold
Z)\:c\equiv 0\bmod N\}$. The points of $X_0(N)$ can be interpreted as pairs
$(E,C)$, where $E$ is a (generalized) elliptic curve and $C$ is a cyclic
subgroup of $E$. There are special points on $X_0(N)$ related to complex
multiplication: let $K$ be a complex quadratic field such that its ring
of integers $O_K$ contains an ideal $I$ satisfying $O_K/\I\cong \bold Z/(N)$.
From the elliptic curve $\bold C/O_K$ one obtains a point $x=(\bold C/O_K,
I^{-1}/O_K)\in X_0(N) (\bold C)$. It can be shown that the point $x$ is
defined over $H$, the Hilbert class field of $K$. Let $E$ be an elliptic
curve over $\bold Q$ admitting a nonconstant morphism $\pi\:X_0(N)\to E$.
We may and do assume that $\pi$ maps the cusp $\infty$ to $0\in E$. The
image $\pi(x)$ of $x$ is called a Heegner point on $E$. It is defined over
$H$. By $y_K$ we denote the point $\roman{Trace}_{H/K}\pi(x)$ and we put
$y=y^{\tau-1}_K$, the "minus"-part of $y_K$. Here, $\tau$ denotes
complex conjugation. The point $y$ is in $E(K)$. In the present paper the following result is proved. Theorem: If $y$ has infinite order in $E(K)$ then the Mordell-Weil group $E(\bold Q)$ and the Tate-Shafarevich group $\text{\cyr{Sh}}(E/\bold Q)$ of $E$ are finite. This result is due to \n V. A. Kolyvagin\en \ref[Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522--540; MR 89m:11056; ibid. Ser. Mat. 52 (1988), no. 6, 1154--1180; MR 90f:11035]. The proof given by the author is a simplified version of Kolyvagin's original proof. One obtains the following important corollary from this. According to the theorem of \n B. H. Gross\en and \n D. B. Zagier\en \ref[Invent. Math. 84 (1986), no. 2, 225--320; MR 87j:11057] one has that the canonical height of $y\in E(K)$ is, up to some trivial nonzero constant, equal to the derivative of the $L$-series of $E$ over $K$ at $1$. Therefore, $y$ has infinite order if and only if $L'(E_{/K},1)=L(E,1)L'(E^{(K)},1)$ is not zero. Here $E^{(K)}$ denotes the twist of $E$ over $K$. It is an elliptic curve over $\bold Q$. By a recent result of \n D. Bump\en, \n S. Friedberg\en and \n J. Hoffstein\en \ref[Bull. Amer. Math. Soc. 21 (1989), no. 1, 89--93; MR 90b:11063] there exists for every elliptic curve $E$ over $\bold Q$ a complex quadratic field $K$ such that $L'(E^{(K)},1)$ is not zero. As a consequence one obtains the following for elliptic curves that admit for some $N$ a nonconstant morphism $\pi\:X_0(N)\to E$: if $L(E,1)\neq 0$ then $E(\bold Q)$ and $\text{\cyr{Sh}}(E/\bold Q)$ are finite. This confirms part of the Birch-Swinnerton-Dyer conjectures. For elliptic curves $E$ with complex multiplication this result was obtained earlier: for $E(\bold Q)$ by \n J. Coates\en and \n A. Wiles\en \ref[Invent. Math. 39 (1977), no. 3, 223--251; MR 57 #3134] and for $\text{\cyr{Sh}}(E/\bold Q)$ by the author \ref[ibid. 89 (1987), no. 3, 527--559; MR 89a:11065]. Kolyvagin's proof exploits properties of Heegner points associated to nonmaximal orders $O_p=\bold Z+pO_K$ for suitable primes $p$. Using Tate's local duality theorems they give rise to annihilators of certain Selmer groups and therefore of the Mordell-Weil and Tate-Shafarevich groups. The annihilators of $\text{\cyr{Sh}}(E/\bold Q)$ thus obtained are consistent with the Birch-Swinnerton-Dyer conjectures. {For the entire collection see MR 90h:14001.} Reviewed by René Schoof |