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An explicit Watson-Ichino formula with CM newforms

Bin Guan

TL;DR

The paper provides an explicit Watson-Ichino identity for the central value $L(\tfrac{1}{2},f\otimes\mathrm{ad}\,g)$ in the CM setting, with $f$ a Hecke--Maass form and $g$ a CM newform. It combines a global formula tying the central L-values to a square of a period against explicit local constants $I'_v$, and a thorough local analysis across induced, special, and supercuspidal representations, including the adjoint lift and Kirillov models. The main contribution is the complete computation of local constants, including archimedean adjustments and the effects of non-squarefree discriminants, establishing a usable, explicit formula applicable to problems such as quantum variance for CM forms. The work extends prior results for dihedral CM cases and non-CM settings, clarifying how local representation types at primes influence the global central-value formula. Overall, it provides a concrete tool for connecting period integrals of CM forms with central L-values in a broad, non-squarefree framework, enabling both theoretical and arithmetic applications.

Abstract

In this paper, we extend the work of Humphries and Khan [HK20] to establish an explicit version of Watson--Ichino formula for $L(1/2,f\otimes\mathrm{ad} g)$, where $f$ is a Hecke--Maass form and $g$ is a CM newform.

An explicit Watson-Ichino formula with CM newforms

TL;DR

The paper provides an explicit Watson-Ichino identity for the central value in the CM setting, with a Hecke--Maass form and a CM newform. It combines a global formula tying the central L-values to a square of a period against explicit local constants , and a thorough local analysis across induced, special, and supercuspidal representations, including the adjoint lift and Kirillov models. The main contribution is the complete computation of local constants, including archimedean adjustments and the effects of non-squarefree discriminants, establishing a usable, explicit formula applicable to problems such as quantum variance for CM forms. The work extends prior results for dihedral CM cases and non-CM settings, clarifying how local representation types at primes influence the global central-value formula. Overall, it provides a concrete tool for connecting period integrals of CM forms with central L-values in a broad, non-squarefree framework, enabling both theoretical and arithmetic applications.

Abstract

In this paper, we extend the work of Humphries and Khan [HK20] to establish an explicit version of Watson--Ichino formula for , where is a Hecke--Maass form and is a CM newform.
Paper Structure (14 sections, 18 theorems, 136 equations)

This paper contains 14 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Organization of the paper
  4. An explicit version of Watson--Ichino Formula
  5. Local calculation in the Watson--Ichino Formula
  6. Whittaker models
  7. Whittaker functions for induced representations
  8. Whittaker functions for supercuspidal representations
  9. The adjoint lift of a supercuspidal representation
  10. Local constants in the Watson--Ichino formula
  11. Proof of Proposition \ref{['localconst.special']}
  12. Proof of Proposition \ref{['localconst.unram']}
  13. Proof of Proposition \ref{['localconst.supercuspidal']}
  14. Direct calculation by matrix coefficients

Key Result

Theorem 1.1

Let $q_1\mid q=|D|$ with $D<0$ a fundamental discriminant, $\mathcal{B}_0^*(q_1)$ be the set of (normalized) Hecke--Maass newforms of weight $0$ and level $q_1$ with trivial nebentypus. For any CM newform $g=g_\Omega\in S_{k}^*(\Gamma_0(q),\chi_D)$ and Hecke--Maass newform $f\in\mathcal{B}_0^*(q_1)$ (notice that $|y^{k/2}g(z)|$ is $\Gamma_0(q)$-invariant), where otherwise, when $4\mid q_1$ and $2

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Proposition 2.2: watson2008rankin
  • Proposition 2.3: cf. humphries2020random
  • Proposition 2.4: cf. humphries2020random
  • Proposition 2.5
  • Lemma 3.1: michel2010subconvexity, humphries2020random
  • ...and 20 more