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A regularized theta lift on the symmetric space of $SL_N$

Romain Branchereau

TL;DR

The article develops a regularized theta lift from harmonic weak Maass forms to differential forms on the SL_N symmetric space, establishing adjointness to the derivative of a cohomological theta lift and expressing torus-periods through derivatives of Hilbert-Eisenstein series. Central to the construction are the Mathai-Quillen kernel and a transgression-based kernel that yield a regularized lift whose periods are computable in terms of modular objects and L-values. By connecting the regularized lift to the derivative of the cohomological lift, the authors link geometrically-defined cycles with Fourier coefficients of Eisenstein-type series, including a detailed torus-period formula involving algebraic constants and L-values. The framework provides a bridge between automorphic geometry on SL_N, Kudla-Millson-type correspondences, and arithmetic information encoded in Hilbert-Eisenstein derivatives, with potential implications for special value formulas and arithmetic geometry.

Abstract

We define a regularized lift from harmonic weak Maass forms of weight $2-N$ to differential forms of degree $N-1$ on the symmetric space $\SL_N(\R)/\SO(N)$, that are smooth outside of certain modular symbols. We show that this lift is adjoint to the derivative of a theta lift. We compute periods of the regularized lift over tori and relate them to Fourier coefficients of Hilbert-Eisenstein series.

A regularized theta lift on the symmetric space of $SL_N$

TL;DR

The article develops a regularized theta lift from harmonic weak Maass forms to differential forms on the SL_N symmetric space, establishing adjointness to the derivative of a cohomological theta lift and expressing torus-periods through derivatives of Hilbert-Eisenstein series. Central to the construction are the Mathai-Quillen kernel and a transgression-based kernel that yield a regularized lift whose periods are computable in terms of modular objects and L-values. By connecting the regularized lift to the derivative of the cohomological lift, the authors link geometrically-defined cycles with Fourier coefficients of Eisenstein-type series, including a detailed torus-period formula involving algebraic constants and L-values. The framework provides a bridge between automorphic geometry on SL_N, Kudla-Millson-type correspondences, and arithmetic information encoded in Hilbert-Eisenstein derivatives, with potential implications for special value formulas and arithmetic geometry.

Abstract

We define a regularized lift from harmonic weak Maass forms of weight to differential forms of degree on the symmetric space , that are smooth outside of certain modular symbols. We show that this lift is adjoint to the derivative of a theta lift. We compute periods of the regularized lift over tori and relate them to Fourier coefficients of Hilbert-Eisenstein series.
Paper Structure (29 sections, 23 theorems, 300 equations)

This paper contains 29 sections, 23 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. A cohomological theta lift
  3. A regularized lift
  4. Periods over tori
  5. Overview
  6. Theta lift on $\mathrm{SL}_N$
  7. Special cycles in the locally symmetric space
  8. Differential forms
  9. Pullback of the Thom form
  10. The transgression form
  11. Weil representation
  12. Example when $N=2$
  13. Theta series
  14. Push-forward
  15. Regular and singular vectors
  16. ...and 14 more sections

Key Result

Theorem 1.1

The right-hand side of expression one converges to a differential form smooth outside of $S_f(\phi)$. Moreover, it is integrable over any smooth chain $c\in C_{N-1}(S_\Gamma)$ that intersects $S_f(\phi)$ transversally.

Theorems & Definitions (46)

  • Example
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.2.1
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • ...and 36 more