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Transactions of the American Mathematical Society

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Mixed modular symbols and the generalized cuspidal $1$-motive


Author: Emmanuel Lecouturier
Journal: Trans. Amer. Math. Soc. 374 (2021), 2823-2872
MSC (2020): Primary 11Fxx
DOI: https://doi.org/10.1090/tran/8310
Published electronically: February 8, 2021
MathSciNet review: 4223035
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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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Additional Information

Emmanuel Lecouturier
Affiliation: Yau Mathematical Sciences Center and Department of mathematics, Tsinghua University, Beijing, People’s Republic of China
MR Author ID: 969451

Received by editor(s): July 20, 2019
Received by editor(s) in revised form: July 6, 2020, July 28, 2020, and August 11, 2020
Published electronically: February 8, 2021
Additional Notes: This research was funded by Tsinghua University and the Yau Mathematical Sciences Center
Article copyright: © Copyright 2021 American Mathematical Society