A Modular Symbol with Values in Cusp Forms

TL;DR

This paper generalizes the construction of modular symbols by Borisov and Gunnells to GL2(Q), producing symbols with values in distributions related to cuspidal power series, revealing new relations among these distributions.

Contribution

It introduces a new modular symbol construction for GL2(Q) with values in distributions on M2(Q), extending previous work and uncovering new relations among distributions.

Findings

Constructed modular symbols with values in distributions on M2(Q)

Revealed new relations (Manin Relations) among distributions

Connected the new construction to previous Eisenstein series-based symbols

Abstract

In [B-G1] and [B-G2], Borisov and Gunnells constructed for each level (N > 1) and for each weight (k > 1) a modular symbol with values in $Sk(\Gamma_1(N))$ using products of Eisenstein series. In this paper we generalize this result by producing a modular symbol (for GL2(Q)!!!) with values in locally constant distributions on M2(Q) taking values in the space of cuspidal power series in two variables (see Definition 5). We can recover the above cited result by restricting to a principal open invariant for the action of $\Gamma_1(N)$ and to the homogeneous degree $k-2$ part of the power series. We should also mention that Colmez [Col] constructs similar distributions (zEis(k; j)). The modification in the definition of such distributions allow us to observe further relations among these distributions (Manin Relations) which in turn makes possible the existence of our construction. In the last section we exhibit some instances of these relations for the full modular group.