Euler Product Asymptotics for L-functions of Elliptic Curves

2023/12/08 by Arshay Sheth, Sheth, Arshay
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2312.05236

openalex publication_date 2023/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∏ _{p ≤ x} (N_p)/(p) ∼ C (log x) ^{rank(E(\mathbb Q))} as x → ∞. 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 ord_s=1 L(E, s)= rank(E(\mathbb Q)). In this paper, we prove that if we let r=ord _s=1L(E, s), then under the assumption of the Riemann Hypothesis for L(E, s), we have that ∏ _{p ≤ x} (N_p)/(p) ∼ 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.

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