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

Source abstract

Let E E be an elliptic curve over Q \mathbb {Q} with discriminant Δ E \Delta _E . For primes p p of good reduction, let N p N_p be the number of points modulo p p and write N p = p + 1 − a p N_p=p+1-a_p . In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies lim x → ∞ 1 log ⁡ x ∑ p ≤ x p ∤ Δ E a p log ⁡ p p = − r + 1 2 , $$\begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E}}}\frac {a_p\log p}{p}=-r+\frac {1}{2}, \end{equation*}$$ where r r is the order of the zero of the L L -function L E ( s ) L_{E}(s) of E E at s = 1 s=1 , which is predicted to be the Mordell-Weil rank of E ( Q ) E(\mathbb {Q}) . We show that if the above limit exits, then the limit equals − r + 1 / 2 -r+1/2 . We also relate this to Nagao’s conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit.