On a Conjecture of Birch and Swinnerton-Dyer

Abstract

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.