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Moment of Derivatives of Quadratic Twists of Modular $L$-Functions

Zijie Zhou

TL;DR

The paper proves an unconditional asymptotic for the mixed second moment of derivatives at the central value of quadratic twists of two distinct modular L-functions, a result previously known only under GRH. By adapting Li–KMotivated techniques to the derivative setting, it combines an approximate functional equation with a refined Poisson summation framework to separate diagonal and off-diagonal contributions. The main term arises from diagonal analysis and is expressed via explicit constants involving $L(1,\mathrm{sym}^2 f)$, $L(1, f\otimes g)$, $L(1,\mathrm{sym}^2 g)$ and a holomorphic Euler product $Z^{*}(0,0)$; off-diagonal terms are shown to be negligible through a delicate sieve and Dirichlet-series bounds. The result has implications for the distribution of ranks in the quadratic twist family of paired elliptic curves, shedding light on BSD-related questions without assuming GRH.

Abstract

We prove an asymptotic for the moment of derivatives of quadratic twists of two distinct modular $L$-functions. This was previously known conditionally on GRH by the work of Ian Petrow.

Moment of Derivatives of Quadratic Twists of Modular $L$-Functions

TL;DR

The paper proves an unconditional asymptotic for the mixed second moment of derivatives at the central value of quadratic twists of two distinct modular L-functions, a result previously known only under GRH. By adapting Li–KMotivated techniques to the derivative setting, it combines an approximate functional equation with a refined Poisson summation framework to separate diagonal and off-diagonal contributions. The main term arises from diagonal analysis and is expressed via explicit constants involving , , and a holomorphic Euler product ; off-diagonal terms are shown to be negligible through a delicate sieve and Dirichlet-series bounds. The result has implications for the distribution of ranks in the quadratic twist family of paired elliptic curves, shedding light on BSD-related questions without assuming GRH.

Abstract

We prove an asymptotic for the moment of derivatives of quadratic twists of two distinct modular -functions. This was previously known conditionally on GRH by the work of Ian Petrow.

Paper Structure

This paper contains 3 sections, 2 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Outline of Proof of Theorem \ref{['thm:One']}
  3. Preliminary Results

Key Result

Theorem 1.1

Let $f,g$ be distinct normalized cuspidal Hecke newforms with trivial central character, odd levels $q_1,q_2$ and even weights $\kappa_1,\kappa_2$ (we let $Q=q_1q_2$). Let $F$ be a smooth, nonnegative function compactly supported on $[1/2,2]$, we have In the above, we have and $C'$ is some explicit constant that depends on $f,g$ and $F$. Here, $L(s,f\otimes g)$ is the Rankin-Selberg convolution

Theorems & Definitions (4)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Lemma 3.1