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 $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ 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 $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x)^r$ 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.