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The Geometric Unitary Kudla Conjecture

Martin Raum

Abstract

We prove that, over an arbitrary imaginary quadratic field, every symmetric formal Fourier-Jacobi series of Hermitian modular forms converges and equals the Fourier-Jacobi expansion of a genuine Hermitian modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties of signature $(p,1)$ is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula of Li-Liu.

The Geometric Unitary Kudla Conjecture

Abstract

We prove that, over an arbitrary imaginary quadratic field, every symmetric formal Fourier-Jacobi series of Hermitian modular forms converges and equals the Fourier-Jacobi expansion of a genuine Hermitian modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties of signature is modular of weight for a Weil representation, establishing the geometric unitary Kudla conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula of Li-Liu.
Paper Structure (23 sections, 50 theorems, 250 equations)

This paper contains 23 sections, 50 theorems, 250 equations.

Table of Contents

  1. Elliptic Jacobi forms
  2. Elliptic modular forms
  3. Basics and definition of Jacobi forms
  4. Reduction to positive definite Jacobi indices
  5. Torsion points of prime denominator
  6. Vanishing order at torsion points
  7. Vanishing bound
  8. Hermitian modular and Jacobi forms
  9. Hermitian modular forms
  10. Hermitian Jacobi forms
  11. Connection to elliptic Jacobi forms
  12. Symmetric formal Fourier--Jacobi series
  13. Convergence of formal Fourier--Jacobi series
  14. Algebraicity over the ring of Hermitian modular forms
  15. Growth at torsion points
  16. ...and 8 more sections

Key Result

Theorem 1

The Kudla generating series $\theta^{\mathrm{Kudla}}_L$ is the Fourier series expansion of a Hermitian modular form of weight $p+1$ and type $\rho^{(g)}_L$. More precisely,

Theorems & Definitions (110)

  • Theorem 1: Unitary geometric Kudla conjecture
  • Corollary 2
  • Remark 1
  • Theorem 3: Automatic convergence
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Proposition 1.4
  • proof
  • ...and 100 more