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

Arshay Sheth

TL;DR

This paper studies Euler-product asymptotics for $L$-functions of elliptic curves by analyzing partial Euler products in the right half of the critical strip under the Riemann Hypothesis for $L(E,s)$. It derives a precise expansion $\prod_{\substack{p\le x}}$ (local Euler factors) $= L(E,s) \exp(-r I_s(x) - R_s(x) + U_s(x) + O(\log x/x^{1/6}))$, with $I_s,R_s,U_s$ defined, and shows that there exists a finite-logarithmic-measure set $S$ such that $\prod_{p\le x} N_p/p \sim C (\log x)^r$ where $r=\mathrm{ord}_{s=1} L(E,s)$ and $C=\dfrac{r!}{L^{(r)}(E,1)} \sqrt{2} e^{r\gamma}$. This yields a robust link between the original OBSD conjecture and the modern BSD conjecture, recovering Goldfeld’s result and providing a partial converse, while leveraging an explicit-formula framework and a Gallagher-based refinement to control error terms. The work supplies a deeper understanding of how partial Euler products encode the rank and L-function behavior, with potential implications for the BSD–OBSD correspondence and related conjectures on Euler products. Overall, the paper advances the analytic study of Euler products for elliptic-curve L-functions and strengthens the connection between invariants predicted by BSD and their Euler-product manifestations.

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.

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

TL;DR

This paper studies Euler-product asymptotics for -functions of elliptic curves by analyzing partial Euler products in the right half of the critical strip under the Riemann Hypothesis for . It derives a precise expansion (local Euler factors) , with defined, and shows that there exists a finite-logarithmic-measure set such that where and . This yields a robust link between the original OBSD conjecture and the modern BSD conjecture, recovering Goldfeld’s result and providing a partial converse, while leveraging an explicit-formula framework and a Gallagher-based refinement to control error terms. The work supplies a deeper understanding of how partial Euler products encode the rank and L-function behavior, with potential implications for the BSD–OBSD correspondence and related conjectures on Euler products. Overall, the paper advances the analytic study of Euler products for elliptic-curve L-functions and strengthens the connection between invariants predicted by BSD and their Euler-product manifestations.

Abstract

Let be an elliptic curve and for each prime , let denote the number of points of modulo . The original version of the Birch and Swinnerton-Dyer conjecture asserts that as . Goldfeld (1982) showed that this conjecture implies both the Riemann Hypothesis for and the modern formulation of the conjecture i.e. that . In this paper, we prove that if we let , then under the assumption of the Riemann Hypothesis for , we have that for all 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 in the right-half of the critical strip.
Paper Structure (6 sections, 18 theorems, 101 equations)

This paper contains 6 sections, 18 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Main results of the paper
  3. Comparison with other work
  4. Partial Euler products in the critical strip
  5. Refinement of the error term for psiE(x)
  6. Applications

Key Result

Theorem 1.3

Let $E/\mathbb Q$ be an elliptic curve. If as $x \to \infty$, then $L(E, s)$ satisfies the Riemann Hypothesis and $\mathop{\mathrm{ord}} \limits _{s=1} L(E, s)= \mathrm{rk}(E)$. Moreover, if we set $r:=\mathop{\mathrm{ord}} \limits _{s=1}L(E,s )$, then where $\gamma$ is Euler's constant and $L^{(r)}(E, s)$ is the $r$-th derivative of $L(E, s)$.

Theorems & Definitions (34)

  • Conjecture 1.1: Birch and Swinnerton-Dyer BirchSwinnertonDyer1965
  • Conjecture 1.2
  • Theorem 1.3: Goldfeld Goldfeld1982
  • Theorem A: Theorem \ref{['mainthm']}
  • Definition 1.4
  • Theorem B: Theorem \ref{['appln']}
  • Corollary A: Corollary \ref{['cor2']}
  • Corollary B: Corollary \ref{['cor1']}
  • Lemma 2.1
  • proof
  • ...and 24 more