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

Abstract

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.