The Birch–swinnerton-dyer Conjecture and Heegner Points: a Survey

For a (connected) smooth projective curve C over the rational numbers Q, it is known that the rational points CpQq depends on the genus g “ gpCq of C: (1) If g “ 0, then the local-global principle holds for C, i.e.: CpQq ‰ H if and only if CpQpq ‰ H for all primes p ď 8 (we understand Qp “ R when p “ 8). In other words, C is globally solvable if and only if it is locally solvable everywhere. Another way of stating this is: C » PQ if and only if CQp » P1Qp for all primes p ď 8. We see that CpQq is either an empty set or an infinite set. (2) If g “ 1, CpQq may be empty, finite or infinite. This article will focus on this case. (3) If g ě 2, Faltings theorem asserts that CpQq is always finite.