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Pullback formula for vector-valued Hermitian modular forms on $\mathrm{U}_{n,n}$

Nobuki Takeda

TL;DR

This work derives a pullback formula for vector-valued Hermitian modular forms on $\mathrm{U}_n$ over CM fields by combining Shimura's finite-place arithmeticity with Kozima's infinite-place differential-operator framework. It develops explicit derivative formulas and pluriharmonic-determined differential operators that preserve automorphy (Condition (A)) via Howe duality, enabling a representation-theoretic treatment of pullbacks. The main result expresses the Petersson pairing of a differentiated Hermitian Eisenstein series with a Hecke eigenform as a product of local $L$-factors and a global pullback of the eigenform, with archimedean and finite-level contributions carefully isolated (via double coset decompositions and local computations). This yields a robust tool for studying Fourier coefficients, automorphic properties, and congruences of vector-valued Hermitian modular forms, with concrete local factors computed at good/bad non-archimedean places and archimedean places.

Abstract

We give the pullback formula for vector-valued Hermitian modular forms on CM field. We also give the equivalent condition for a differential operator on Hermitian modular forms to preserve the automorphic properties.

Pullback formula for vector-valued Hermitian modular forms on $\mathrm{U}_{n,n}$

TL;DR

This work derives a pullback formula for vector-valued Hermitian modular forms on over CM fields by combining Shimura's finite-place arithmeticity with Kozima's infinite-place differential-operator framework. It develops explicit derivative formulas and pluriharmonic-determined differential operators that preserve automorphy (Condition (A)) via Howe duality, enabling a representation-theoretic treatment of pullbacks. The main result expresses the Petersson pairing of a differentiated Hermitian Eisenstein series with a Hecke eigenform as a product of local -factors and a global pullback of the eigenform, with archimedean and finite-level contributions carefully isolated (via double coset decompositions and local computations). This yields a robust tool for studying Fourier coefficients, automorphic properties, and congruences of vector-valued Hermitian modular forms, with concrete local factors computed at good/bad non-archimedean places and archimedean places.

Abstract

We give the pullback formula for vector-valued Hermitian modular forms on CM field. We also give the equivalent condition for a differential operator on Hermitian modular forms to preserve the automorphic properties.

Paper Structure

This paper contains 16 sections, 27 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Hermitian Modular Forms
  3. As Analytic Functions on hermitian Symmetric Spaces
  4. As Functions on $\mathrm{U}(n,n)$.
  5. As Functions on Unitary Groups over the Adeles.
  6. Differential Operators
  7. Formulas on Derivatives
  8. Differential Operators on Automorphic Forms
  9. Hermitian Eisenstein Series
  10. Pullback Formula
  11. Double Coset Decomposition
  12. Pullback Formula
  13. Local Computations
  14. Good Non-archimedean Factors
  15. Bad Non-archimedean Factors
  16. ...and 1 more sections

Key Result

Proposition 2.4

The above correspondence $f\mapsto \phi_f$ gives the isomorphism

Theorems & Definitions (54)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: e.g. Eischen2024Automorphic
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 3.1
  • proof
  • ...and 44 more