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$L$-derivatives of the fixed elliptic curve over rank-one imaginary quadratic fields

Shenghao Hua

TL;DR

This work studies the distribution of logarithms of derivatives of base-change L-functions L'(1, E/ Q(√d)) for a fixed non-CM elliptic curve E over Q, as d runs over negative fundamental discriminants with specified congruence conditions. It develops a twisted first-moment framework and employs a Radziwiłł–Soundararajan–style approach to establish a one-sided central limit theorem for log L'(1, E/ Q(√d)) under rank-one constraints, with conditional extensions under GRH. It also shows that many twists yield large L'-values and provides precise asymptotics and bounds for related twisted sums, including a refined Proposition with an extra logarithmic factor. The results deepen understanding of L'-value distributions in quadratic base changes and have implications for the arithmetic of E over imaginary quadratic fields, within the Langlands program and potential GRH assumptions.

Abstract

There is a one-sided central limit theorem for the logarithms of $L$-derivatives of a fixed rational non-CM elliptic curve $E$ over imaginary quadratic fields of rank one, analogous to a result of Radziwiłł and Soundararajan. There are also many $L$-derivatives that are not small.

$L$-derivatives of the fixed elliptic curve over rank-one imaginary quadratic fields

TL;DR

This work studies the distribution of logarithms of derivatives of base-change L-functions L'(1, E/ Q(√d)) for a fixed non-CM elliptic curve E over Q, as d runs over negative fundamental discriminants with specified congruence conditions. It develops a twisted first-moment framework and employs a Radziwiłł–Soundararajan–style approach to establish a one-sided central limit theorem for log L'(1, E/ Q(√d)) under rank-one constraints, with conditional extensions under GRH. It also shows that many twists yield large L'-values and provides precise asymptotics and bounds for related twisted sums, including a refined Proposition with an extra logarithmic factor. The results deepen understanding of L'-value distributions in quadratic base changes and have implications for the arithmetic of E over imaginary quadratic fields, within the Langlands program and potential GRH assumptions.

Abstract

There is a one-sided central limit theorem for the logarithms of -derivatives of a fixed rational non-CM elliptic curve over imaginary quadratic fields of rank one, analogous to a result of Radziwiłł and Soundararajan. There are also many -derivatives that are not small.

Paper Structure

This paper contains 4 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preparation
  3. Proof of the Theorems for L-functions
  4. Proof of Proposition \ref{['prop:bound']}

Key Result

Theorem 1.1

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ with conductor $N$ and root number $w$, and the vanishing order of the $L(s,E)$ at the central point is at most 1. Fix a residue class $a \bmod N_0$ be a residue class such that $a$ is a quadratic residue modulo $N$, $a \equiv 1$ or $5 \pmod{8}$, is at most Moreover, when $L(1,E)\neq 0$, this result also applies to $L'(1, E^{(d)})$.

Theorems & Definitions (14)

  • Theorem 1.1: One-sided central limit theorem
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3: RadziwillSoundararajan2015, Lemma 3
  • Lemma 2.4
  • proof
  • Lemma 2.5: RadziwillSoundararajan2015, Lemma 1
  • ...and 4 more