From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture

Seoyoung Kim, M. Ram Murty

Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant ${\Delta }_E$. For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$ and write $N_p=p+1-a_p$. In 1…