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A $Λ$-adic Kudla lift

Francesco Maria Iudica

TL;DR

This work develops a $p$-adic interpolation framework for Kudla lifts from elliptic modular forms to Picard modular forms on GU(2,1). By introducing a level $p$ modification and leveraging Finis’ explicit Fourier–Jacobi formulas, the authors construct a $oldsymbol{\Lambda}$-adic lift that varies coherently with weight and level along Hida families. They establish $p$-adic Hecke equivariance, interpolate a family of arithmetic theta data, and connect elliptic and Picard sides through $oldsymbol{ extbf{L}}^*$-type lifts, culminating in a $oldsymbol{\\Lambda}$-adic lift whose specializations reproduce the classical Kudla lift at arithmetic weights. The results provide a concrete, p-adically continuous bridge between automorphic theta correspondences for unitary groups and $oldsymbol{\\Lambda}$-adic families, with potential applications to $p$-adic families of special cycles and adjoint lifts. Overall, the paper delivers explicit constructions and interpolation techniques that advance the understanding of $p$-adic variation in arithmetic theta lifts for Picard modular forms.

Abstract

The Kudla lift studied in this article is a classical version for Picard modular forms of the automorphic theta lift between $\text{GU}(2)$ and $\text{GU}(3)$. We construct an explicit $p$-adic analytic family of Picard modular forms varying with respect to the weight and level, which interpolates a so-called $p$-modification of the lift at arithmetic weights, by exploiting a formula of Finis for the Fourier-Jacobi coefficients of a lifted form.

A $Λ$-adic Kudla lift

TL;DR

This work develops a -adic interpolation framework for Kudla lifts from elliptic modular forms to Picard modular forms on GU(2,1). By introducing a level modification and leveraging Finis’ explicit Fourier–Jacobi formulas, the authors construct a -adic lift that varies coherently with weight and level along Hida families. They establish -adic Hecke equivariance, interpolate a family of arithmetic theta data, and connect elliptic and Picard sides through -type lifts, culminating in a -adic lift whose specializations reproduce the classical Kudla lift at arithmetic weights. The results provide a concrete, p-adically continuous bridge between automorphic theta correspondences for unitary groups and -adic families, with potential applications to -adic families of special cycles and adjoint lifts. Overall, the paper delivers explicit constructions and interpolation techniques that advance the understanding of -adic variation in arithmetic theta lifts for Picard modular forms.

Abstract

The Kudla lift studied in this article is a classical version for Picard modular forms of the automorphic theta lift between and . We construct an explicit -adic analytic family of Picard modular forms varying with respect to the weight and level, which interpolates a so-called -modification of the lift at arithmetic weights, by exploiting a formula of Finis for the Fourier-Jacobi coefficients of a lifted form.

Paper Structure

This paper contains 16 sections, 23 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Conventions on Hecke characters
  4. The unitary Shimura variety of signature $(2,1)$
  5. PEL type moduli spaces
  6. Picard modular forms
  7. The Fourier-Jacobi expansion
  8. Arithmetic theta functions
  9. The $p$-modified Kudla lift
  10. The level one Kudla lift
  11. The $p$-modification
  12. The $p$-adic interpolation of the Kudla lift
  13. On $\Lambda$-adic elliptic cusp forms
  14. On $\Lambda$-adic Picard modular forms
  15. The $p$-adic lift
  16. ...and 1 more sections

Key Result

Theorem 1

If the level $\mathcal{U}^p$ is neat, the above moduli problem is represented by a smooth, quasi-projective scheme $S_{\mathcal{U}_f}$ over $\mathcal{O}_{K,\mathfrak{p}}$.

Theorems & Definitions (57)

  • Definition 1
  • Theorem 1
  • Definition 2: Picard modular form
  • Remark 1
  • Theorem 2
  • proof
  • Definition 3
  • Theorem 3
  • Proposition 1
  • proof
  • ...and 47 more