  For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture
predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of
the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for
elliptic curves with complex multiplications. The key method of the proof is to
reduce the Galois action of infinite order on the Tate module of an elliptic
curve to that of finite order by using the $p$-adic Hodge theory. As a
corollary, we can determine whether a given natural number is a congruent
number (congruent number problem). This problem is one of the oldest unsolved
problems in mathematics.
