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

Abstract

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 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies $$\underset{x\to \infty }{\lim }\frac{1}{\log x}\sum _{\begin{array}{c}p\le x\\ p\nmid {\Delta }_E\end{array}}\frac{a_p\log p}{p}=-r+\frac{1}{2},$$ where $r$ is the order of the zero of the $L$-function $L_E(s)$ of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of $E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals $-r+1/2$. We also relate this to Nagao's conjecture.

23 pages, with an appendix by Andrew V. Sutherland