From the Birch and Swinnerton-Dyer conjecture to Nagao’s conjecture
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- by Seoyoung Kim and M. Ram Murty; with an appendix by Andrew V. Sutherland
- Math. Comp. 92 (2023), 385-408
- DOI: https://doi.org/10.1090/mcom/3773
- Published electronically: September 12, 2022
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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 \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 $r$ 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. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit.References
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Bibliographic Information
- Seoyoung Kim
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- MR Author ID: 1343309
- ORCID: 0000-0003-1759-6480
- Email: sk206@queensu.ca
- M. Ram Murty
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- MR Author ID: 128555
- Email: murty@queensu.ca
- Andrew V. Sutherland
- MR Author ID: 852273
- ORCID: 0000-0001-7739-2792
- Received by editor(s): November 6, 2021
- Received by editor(s) in revised form: May 19, 2022, and June 12, 2022
- Published electronically: September 12, 2022
- Additional Notes: The first author was partially supported by a Coleman Postdoctoral Fellowship. The second author was partially supported by NSERC Discovery grant
Dedicated to the memory of Professor John H. Coates - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 385-408
- MSC (2020): Primary 11G40; Secondary 14G10, 14D10, 14H52
- DOI: https://doi.org/10.1090/mcom/3773
- MathSciNet review: 4496969