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Test Schwartz functions at $p$ for theta lifts of Hida families

Zheng Liu

TL;DR

The work constructs explicit $p$-adic families of theta lifts from definite orthogonal and unitary groups to symplectic/quasi-split unitary targets by carefully selecting test Schwartz functions at places above $p$ with $\,\mathbb{U}_p$-equivariance. It develops a local equivariance framework, uses the Weil representation to interpolate Fourier coefficients, and integrates $p$-adic Petersson theory with Hida theory to produce $ ext{I}$-adic, $P$-ordinary theta lifts from $H$ to $G'$ (and hence to $G$). A central achievement is the explicit specialization formula for the lifts and the construction of measure-valued theta data that interpolate the lifts across finite-order characters, culminating in an $ ext{I}$-adic $P$-ordinary family of theta lifts. The results provide a concrete mechanism to vary theta-lift data in $p$-adic families, enabling new avenues for studying $p$-adic $L$-functions and Galois representations attached to higher-rank automorphic forms.

Abstract

We construct Hida families of theta lifts from definite orthogonal and unitary groups. A major ingredient of the construction is the choice of test Schwartz functions at places dividing $p$. We select a special type of Schwartz functions endowed with the equivariance property for the action of $\mathbb{U}_p$-operators.

Test Schwartz functions at $p$ for theta lifts of Hida families

TL;DR

The work constructs explicit -adic families of theta lifts from definite orthogonal and unitary groups to symplectic/quasi-split unitary targets by carefully selecting test Schwartz functions at places above with -equivariance. It develops a local equivariance framework, uses the Weil representation to interpolate Fourier coefficients, and integrates -adic Petersson theory with Hida theory to produce -adic, -ordinary theta lifts from to (and hence to ). A central achievement is the explicit specialization formula for the lifts and the construction of measure-valued theta data that interpolate the lifts across finite-order characters, culminating in an -adic -ordinary family of theta lifts. The results provide a concrete mechanism to vary theta-lift data in -adic families, enabling new avenues for studying -adic -functions and Galois representations attached to higher-rank automorphic forms.

Abstract

We construct Hida families of theta lifts from definite orthogonal and unitary groups. A major ingredient of the construction is the choice of test Schwartz functions at places dividing . We select a special type of Schwartz functions endowed with the equivariance property for the action of -operators.
Paper Structure (20 sections, 7 theorems, 111 equations)

This paper contains 20 sections, 7 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. $\mathbb{U}_\mathfrak{p}$-equivariant test Schwartz functions at $\mathfrak{p}$
  3. The setting and notation
  4. Some formulas for Weil representations
  5. The $\mathbb{U}_p$-operators
  6. A special type of test Schwartz functions
  7. The $\mathbb{U}_p$-equivariance
  8. Theta lifts of Hida families on definite orthogonal and unitary groups
  9. Some setting
  10. $p$-adic families of automorphic forms on $H$ and Petersson inner product
  11. $p$-adic families on $H$
  12. A $p$-adic Petersson inner product
  13. Constructing $p$-adic families of $q$-expansions on $G$
  14. Interpolating Fourier coefficients of theta series
  15. Some more setting
  16. ...and 5 more sections

Key Result

Theorem 1.0.1

The Schwartz function $\phi_{\underline{\chi}_\mathfrak{p}}$ defined in eq:Schwchi satisfies the following equivariance property for the $U_p$-operators defined in eq:UGpteq:UHpt: For ${\tt t}= \mathrm{diag}(a_1,\dots,a_m,\bar{a}^{-1}_1,\dots,\bar{a}^{-1}_m)\in H_\mathfrak{p}$ satisfying the conditi Here $A(\breve{{\tt t}})=\mathrm{diag}(a_1,\dots,a_m,1,\dots,1)\in\mathop{\mathrm{GL}}\nolimits_n(K

Theorems & Definitions (13)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 2.5.1
  • proof
  • Proposition 3.2.1
  • proof
  • Proposition 3.3.1
  • proof
  • Proposition 3.3.2
  • proof
  • ...and 3 more