Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-08T04:30:10.191Z Has data issue: false hasContentIssue false

On a Conjecture of Birch and Swinnerton-Dyer

Published online by Cambridge University Press:  20 November 2018

Wentang Kuo
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, e-mail: wtkuo@math.uwaterloo.ca
M. Ram Murty
Affiliation:
Department of Mathematics and Statistics, Queens University, Kingston, ON, K7L 3N6, e-mail: murty@mast.queensu.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $E/\mathbb{Q}$ be an elliptic curve defined by the equation ${{y}^{2}}\,=\,{{x}^{3}}\,+\,ax\,+\,b.$ For a prime $p$, $p\nmid \Delta \,=\,-16\left( 4{{a}^{3}}+27{{b}^{2}} \right)\,\ne \,0,$ define

$${{N}_{p}}=p+1-{{a}_{p}}=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|.$$

As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$,

$$\prod\limits_{p\le x,p\nmid \Delta }{\frac{{{N}_{p}}}{p}\,\sim \,c{{\left( \log x \right)}^{r}},\,\,\,x\to \infty .}$$

Let ${{\alpha }_{p}}$ and ${{\beta }_{p}}$ be the eigenvalues of the Frobenius at $p$. Define

$${{\tilde{c}}_{n}}=\left\{ \begin{align} & \frac{\alpha _{p}^{k}+\beta _{P}^{k}}{k}\,\,\,\,n={{p}^{k}},\,p\,\text{is}\,\text{a}\,\text{prime,}\,k\,\text{is}\,\text{a}\,\text{natural}\,\text{number},\,p\nmid \Delta . \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\ \end{align} \right.\,$$

and $\tilde{C}\left( x \right)\,=\,\sum\nolimits_{n\le x}{{{{\tilde{c}}}_{n}}.}$ In this paper, we establish the equivalence between the conjecture and the condition $\tilde{C}\left( x \right)\,=\,\mathbf{o}\left( x \right).$ The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an $\Omega $ theorem which relate to the constant $c$ in the conjecture.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Bateman, P. T. and Diamond, H. G., On the oscillation theorems of Pringsheim and Landau. In: Number theory, Trends Math, Birkhäuser, Basel, 2000, pp. 4354,Google Scholar
[2] Birch, B., Conjectures concerning elliptic curves. Proc. Symp. Pure Math, 8, Amer. Math. Soc, Providence, RI, 1965, pp. 106112.Google Scholar
[3] Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over Q: wild 3-adic exercises. Amer, J.. Math. Soc. 14(2001), 843939.Google Scholar
[4] Conrad, K., Partial Euler products on the critical line. Canad. Math. J. 57(2005), 267297.Google Scholar
[5] Goldfeld, D., Sur les produits eulériens attachés aux courbes elliptiques. Acad, C. R.. Sci. Paris Sér. I Math. 294(1982), 471474.Google Scholar
[6] Hans Arnold, Heilbronn, The collected papers of Hans Arnold Heilbronn. JohnWiley and Sons, New York, 1988, pp. 168174.Google Scholar
[7] Iwaniec, H., Topics in classical automorphic forms. Graduate Studies in Mathematics, 17, American Math. Soc., Providence, RI, 1997.Google Scholar
[8] Montgomery, H. L., The zeta function and prime numbers. Queen's Papers in Pure and Appl.Math. 54(1980), 131.Google Scholar
[9] Riesz, M., Ein Konvergenzsatz für Dirichletsche Reihen. Acta Mathematica 40(1916), 350354.Google Scholar
[10] Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141(1995), 443551.Google Scholar