Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Explicit Theta Lift to Siegel Paramodular Forms

Jennifer Johnson-Leung, Nina Rupert

TL;DR

The paper develops an explicit local theta lift from ${ m GO}(X)$ to ${ m GSp}(4)$ via the Weil representation to realize Siegel paramodular forms as lifts of Hilbert cusp forms on ${ m GL}(2,E)$ over real quadratic fields. It provides a complete local recipe: for each non-wildly ramified local type (split, inert, or tamely ramified), it constructs a Schwartz function $varphi$ and an input Whittaker datum $W$ so that the local lift $B(g,varphi,W,s)$ is nonzero and ${ m K}({ rak p}^N)$-invariant, with $N$ determined by the local type. The results yield explicit paramodular vectors at all finite places, align with the global theta lift, and supply a concrete pathway to compute Fourier coefficients of the resulting paramodular forms, offering evidence for the Brumer–Kramer paramodular conjecture in this setting. Wild ramification remains outside the construction, and archimedean aspects are treated with analytic refinements. Overall, the work advances the explicit realization of the theta correspondence in the paramodular context and provides practical tools for constructing and analyzing Siegel modular lifts from real-quadratic Hilbert cusp forms.

Abstract

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $Γ_0$ level to an irreducible automorphic representation of $\mathrm{GSp}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.

An Explicit Theta Lift to Siegel Paramodular Forms

TL;DR

The paper develops an explicit local theta lift from to via the Weil representation to realize Siegel paramodular forms as lifts of Hilbert cusp forms on over real quadratic fields. It provides a complete local recipe: for each non-wildly ramified local type (split, inert, or tamely ramified), it constructs a Schwartz function and an input Whittaker datum so that the local lift is nonzero and -invariant, with determined by the local type. The results yield explicit paramodular vectors at all finite places, align with the global theta lift, and supply a concrete pathway to compute Fourier coefficients of the resulting paramodular forms, offering evidence for the Brumer–Kramer paramodular conjecture in this setting. Wild ramification remains outside the construction, and archimedean aspects are treated with analytic refinements. Overall, the work advances the explicit realization of the theta correspondence in the paramodular context and provides practical tools for constructing and analyzing Siegel modular lifts from real-quadratic Hilbert cusp forms.

Abstract

Let be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of which contains a Hilbert modular form with level to an irreducible automorphic representation of which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.
Paper Structure (15 sections, 43 theorems, 137 equations, 1 table)

This paper contains 15 sections, 43 theorems, 137 equations, 1 table.

Table of Contents

  1. Introduction
  2. Statement of the Main Theorem
  3. Preliminaries
  4. Whittaker Models
  5. The split case
  6. The non-split case
  7. Additional Results
  8. Weil Representations
  9. Intertwining Maps
  10. The non-Archimedean Case
  11. The Archimedean Case
  12. The explicit Schwartz functions
  13. The Split Case
  14. The Inert Case
  15. The Ramified Case

Key Result

Theorem 3.1

Let $L$ be a non-archimedean local field and let $\psi$ be a non-trivial additive character of $L$. Let $(\tau , V)$ be an irreducible admissible representation of ${\rm GL}(2,L)$. Then $\tau$ is infinite-dimensional if and only if $\tau$ admits a unique Whittaker model.

Theorems & Definitions (84)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • ...and 74 more