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

TL;DR

This paper explores the relationship between the Birch and Swinnerton-Dyer conjecture and Nagao's conjecture, showing that a certain limit related to elliptic curves equals .5 minus the rank if it exists.

Contribution

It proves that if a specific limit involving elliptic curve data exists, then it equals .5 minus the Mordell-Weil rank, connecting BSD and Nagao's conjecture.

Findings

If the limit exists, it equals .5 minus the rank.

The result links the limit to the order of zero of the L-function.

Provides a conditional equivalence between the conjectures.

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 $$\lim_{x\to\infty}\frac{1}{\log x}\sum_{\substack{p\leq x\\ p\nmid \Delta_{E}}}\frac{a_p\log p}{p}=-r+\frac{1}{2},$$ 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.