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The Second Moment of Rankin-Selberg $L$-Functions in Conductor-Dropping Regimes

Peter Humphries, Rizwanur Khan

TL;DR

This work addresses the subconvexity challenge for adjoint L-functions by studying the second moment of Rankin–Selberg L-functions $L(s, f \otimes g)$ at the central point with conductor-dropping inputs. The authors construct an exact identity by evaluating a period integral in two spectral ways, yielding explicit main-term contributions and manageable error terms. They derive an asymptotic formula for the second moment over holomorphic cusp forms of fixed weight, with a main term of the form $k \sum_{j=0}^{3} P_j(\log k) L^{(3-j)}(1, \mathrm{ad}\, g)$ and an error $O_{\varepsilon}(k (\log k)^{-1/2+\varepsilon})$, and prove positivity and size bounds for the main term. Additionally, they discuss generalisations to level and Maaß settings, bound the weight functions that appear in the spectral expansions, and provide a detailed analysis of the main term via residues and Dirichlet-series techniques. The results illuminate conductor-dropping phenomena and offer a pathway toward subconvexity via amplification, while clarifying structural features of the main term that persist beyond the central point.

Abstract

We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.

The Second Moment of Rankin-Selberg $L$-Functions in Conductor-Dropping Regimes

TL;DR

This work addresses the subconvexity challenge for adjoint L-functions by studying the second moment of Rankin–Selberg L-functions at the central point with conductor-dropping inputs. The authors construct an exact identity by evaluating a period integral in two spectral ways, yielding explicit main-term contributions and manageable error terms. They derive an asymptotic formula for the second moment over holomorphic cusp forms of fixed weight, with a main term of the form and an error , and prove positivity and size bounds for the main term. Additionally, they discuss generalisations to level and Maaß settings, bound the weight functions that appear in the spectral expansions, and provide a detailed analysis of the main term via residues and Dirichlet-series techniques. The results illuminate conductor-dropping phenomena and offer a pathway toward subconvexity via amplification, while clarifying structural features of the main term that persist beyond the central point.

Abstract

We prove an asymptotic formula for the second moment of -functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.
Paper Structure (10 sections, 27 theorems, 129 equations)

This paper contains 10 sections, 27 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Generalisations
  4. An Exact Identity for Moments of L-Functions
  5. The First Spectral Expansion
  6. The Second Spectral Expansion
  7. Bounds for Moments of L-Functions
  8. The Size of the Weight Functions
  9. The Size of Moments of L-Functions
  10. Bounds for the Main Term

Key Result

Theorem 1.4

Let $\mathcal{B}_{\mathrm{hol}}$ denote an orthonormal basis of holomorphic Hecke cusp forms $f$ of positive even weight $k_f \in 2\mathbb{N}$. Let $g \in \mathcal{B}_{\mathrm{hol}}$ be a holomorphic Hecke cusp form of positive even weight $k \in 2\mathbb{N}$. For $j \in \{0,1,2,3\}$, there exist po for any $\varepsilon > 0$. Moreover, the main term $k \sum_{j = 0}^{3} P_{j}(\log k) L^{(3 - j)}(1,

Theorems & Definitions (52)

  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.17
  • Lemma 2.19
  • proof
  • Lemma 2.21
  • proof
  • Lemma 2.24
  • proof
  • Lemma 2.28
  • ...and 42 more