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

TL;DR

This paper introduces a new theta lift connecting homology of locally symmetric spaces for SL_N with modular forms, revealing explicit Fourier coefficients linked to modular symbols and cohomology classes, especially for N=2.

Contribution

It defines a novel theta lift generalizing Kudla-Millson in the SL_N setting, with explicit Fourier coefficients and surjectivity results for N=2.

Findings

Fourier coefficients are Poincaré duals of modular symbols

Constant term corresponds to transgressed Euler class

Surjectivity onto certain weight 2 modular forms when N=2

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 $N$, similar to the Kudla-Millson lift in the orthogonal setting. We show that the Fourier coefficients of this lift are Poincar\'e duals of 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. 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.