Eisenstein classes and generating series of modular symbols in SL_N

2024/11/13 by Branchereau, Romain
#11F03 #11F11 #11F23 #11F27 #11F30 #11F67 #53C07 #53C22 #53C30 #55N45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.08690

Abstract

We define a theta lift between the homology in degree N-1 of a locally symmetric space associated to SL_N(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 SL_N × SL₂, 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.

Related

• A regularized theta lift on the symmetric space of SL_N
• Theta correspondence and the Borisov-Gunnells relations
• Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups
• Lefschetz decompositions of Kudla-Millson theta functions
• Eisenstein cycles as modular symbols