Euler Product Asymptotics for $L$-functions of Elliptic Curves

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the Birch and Swinnerton-Dyer conjecture asserts that $\prod _{p\le x}\frac{N_p}{p}\sim C(\log x{)}^{\text{rank}(E(\mathbb{Q}))}$ as $x\to \infty$. Goldfeld (1982) showed that this conjecture implies both the Riemann Hypothesis for $L(E,s)$ and the modern formulation of the conjecture i.e. that ${\text{ord}}_{s=1}L(E,s)=\text{rank}(E(\mathbb{Q}))$. In this paper, we prove that if we let $r={\text{ord}}_{s=1}L(E,s)$, then under the assumption of the Riemann Hypothesis for $L(E,s)$, we have that $\prod _{p\le x}\frac{N_p}{p}\sim C(\log x{)}^r$ for all $x$ outside a set of finite logarithmic measure. As corollaries, we recover not only Goldfeld's result, but we also prove a result in the direction of the converse. Our method of proof is based on establishing the asymptotic behaviour of partial Euler products of $L(E,s)$ in the right-half of the critical strip.