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Eisenstein classes and generating series of modular symbols in $\mathrm{SL}_N$

Romain Branchereau

TL;DR

This work constructs a geometric theta lift for the dual pair $\mathrm{SL}_N(\mathbb{R})\times\mathrm{SL}_2(\mathbb{R})$, relating the degree $N-1$ homology of SL$_N$-locally symmetric spaces to weight $N$ modular forms via an Eisenstein class obtained from the Mathai–Quillen formalism. The lift's Fourier coefficients are identified with Poincaré duals to modular symbols $Z_n(\chi)$ attached to maximal parabolics, while the constant term is a canonical Eisenstein class arising from transgressing the Euler class. In the $N=2$ case the lift surjects onto the weight-$2$ space spanned by Eisenstein series and cusp forms with nonzero $L$-values, and the construction recovers a diagonal Hilbert-Eisenstein generating series as in related works. Overall, the paper provides a geometric theta correspondence for $\mathrm{SL}_N\times\mathrm{SL}_2$, connecting modular symbols to Eisenstein data and offering a new perspective on the generating series of modular symbols via automorphic cohomology.

Abstract

We define a theta lift between the homology in degree $N-1$ of a locally symmetric space associated to $\mathrm{SL}_N(\mathbb{R})$ and the space of modular forms of weight $N$. We show that the Fourier coefficients of this lift are Poincaré duals to modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. This Eisenstein lift realizes a geometric theta correspondence for the pair $\mathrm{SL}_N \times \mathrm{SL}_2$, in the spirit of Kudla-Millson. When $N=2$, we show that the lift surjects on the space of weight $2$ modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing $L$-function.

Eisenstein classes and generating series of modular symbols in $\mathrm{SL}_N$

TL;DR

This work constructs a geometric theta lift for the dual pair , relating the degree homology of SL-locally symmetric spaces to weight modular forms via an Eisenstein class obtained from the Mathai–Quillen formalism. The lift's Fourier coefficients are identified with Poincaré duals to modular symbols attached to maximal parabolics, while the constant term is a canonical Eisenstein class arising from transgressing the Euler class. In the case the lift surjects onto the weight- space spanned by Eisenstein series and cusp forms with nonzero -values, and the construction recovers a diagonal Hilbert-Eisenstein generating series as in related works. Overall, the paper provides a geometric theta correspondence for , connecting modular symbols to Eisenstein data and offering a new perspective on the generating series of modular symbols via automorphic cohomology.

Abstract

We define a theta lift between the homology in degree of a locally symmetric space associated to and the space of modular forms of weight . We show that the Fourier coefficients of this lift are Poincaré duals to modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. This Eisenstein lift realizes a geometric theta correspondence for the pair , in the spirit of Kudla-Millson. When , we show that the lift surjects on the space of weight modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing -function.

Paper Structure

This paper contains 51 sections, 50 theorems, 465 equations.

Table of Contents

  1. Introduction
  2. The work of Kudla-Millson
  3. Special cycles for $\mathrm{SL}_N(\mathbb{R})$
  4. The Eisenstein class
  5. Theta lifts
  6. Evaluation on tori
  7. The case of the modular curve
  8. Outline of the paper
  9. Acknowledgements
  10. Locally symmetric spaces and special cycles
  11. Locally symmetric space of $\mathrm{SL}_N(\mathbb{R})$
  12. Special cycles in $S$
  13. Quotients
  14. Special cycles in the quotient.
  15. Orientability.
  16. ...and 36 more sections

Key Result

Theorem 1.1

The pushforward integral112 converges absolutely for any $s \in \mathbb{C}$. At $s=0$ it defines a closed $(N-1)$-form $E_{\varphi\chi}(z_1,\tau)$, which is invariant by $\Gamma \subset \mathrm{SL}_N(\mathbb{R})$ and transforms in $\tau$ like a modular form of weight $N$ and level $\Gamma' \subset \ where $[Z_n(\chi)]$ denotes the Poincaré dual to $Z_n(\chi)$.

Theorems & Definitions (105)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.4
  • Corollary 1.5.1
  • ...and 95 more