# Mixed modular symbols and the generalized cuspidal 1-motive

## TL;DR

This paper introduces mixed modular symbols, extending classical modular symbols to encode more information about Eisenstein series, and constructs related 1-motives connected to the Jacobian of modular curves.

## Contribution

It defines and studies the space of mixed modular symbols, linking them to 1-motives and generalized Jacobians, and relates the construction to p-adic periods for specific subgroups.

## Key findings

- Mixed modular symbols extend classical symbols and capture more Eisenstein series information.
- Construction of 1-motives related to the generalized Jacobian of modular curves.
- Connection to p-adic periods of modular curves for specific subgroups.

## Abstract

We define and study the space of mixed modular symbols for a given finite index subgroup $\Gamma$ of $SL_2(\mathbf{Z})$. This is an extension of the usual space of modular symbols, which in some cases carries more information about Eisenstein series. We make use of mixed modular symbols to construct some $1$-motives related to the generalized Jacobian of modular curves. In the case $\Gamma = \Gamma_0(p)$ for some prime $p$, we relate our construction to the work of Ehud de Shalit on $p$-adic periods of $X_0(p)$.