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

Birch and Swinnerton-Dyer conjecture in the complex multiplication case and the congruent number problem