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

Let E E be an elliptic curve over alttext="double-struck upper Q"> Q encoding="application/x-tex">\mathbb {Q} with discriminant alttext="normal Delta Subscript mathvariant="normal">Δ encoding="application/x-tex">\Delta _E . For primes alttext="p"> p encoding="application/x-tex">p of good reduction, let N p"> N encoding="application/x-tex">N_p the number points modulo and write p Baseline equals plus 1 minus a = + 1 − a encoding="application/x-tex">N_p=p+1-a_p In 1965, Birch Swinnerton-Dyer formulated conjecture which implies lim x stretchy="false">→<!-- → mathvariant="normal">∞<!-- ∞ </mml:munder> log ⁡<!-- ⁡ </mml:mfrac> ∑<!-- ∑ <mml:mstyle scriptlevel="1"> ≤<!-- ≤ </mml:mtd> ∤<!-- ∤ </mml:mtable> r 2 , encoding="application/x-tex">\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 alttext="r"> encoding="application/x-tex">r is order zero L"> L encoding="application/x-tex">L -function L left-parenthesis s right-parenthesis"> stretchy="false">( s stretchy="false">) encoding="application/x-tex">L_{E}(s) at alttext="s 1"> encoding="application/x-tex">s=1 , predicted to Mordell-Weil rank double-struck Q encoding="application/x-tex">E(\mathbb {Q}) We show that if above limit exits, then alttext="negative slash 2"> / encoding="application/x-tex">-r+1/2 also relate this Nagao’s conjecture. This paper includes appendix by Andrew V. Sutherland gives evidence for convergence above-mentioned limit.