Mixed modular symbols and the generalized cuspidal 1-motive

Lecouturier, Emmanuel

doi:10.1090/tran/8310

Mixed modular symbols and the generalized cuspidal $1$-motive
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Trans. Amer. Math. Soc. 374 (2021), 2823-2872 Request permission

Abstract:

We define and study the space of mixed modular symbols for a given finite index subgroup $\Gamma$ of $\operatorname {SL}_2(\mathbf {Z})$. This is an extension of the usual space of modular symbols that, 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 $X_0(p)$.

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