The Eisenstein cycles as modular symbols

For any odd integer N, we explicitly write down the Eisenstein cycles in the first homology group of modular curves of level N as linear combinations of Manin symbols. These cycles are, by definition, those over which every integral of holomorphic differential forms vanish. Our result can be seen as...

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Published in:Journal of the London Mathematical Society 2018-10, Vol.98 (2), p.329-348
Main Authors: Banerjee, Debargha, Merel, Loïc
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Language:English
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description For any odd integer N, we explicitly write down the Eisenstein cycles in the first homology group of modular curves of level N as linear combinations of Manin symbols. These cycles are, by definition, those over which every integral of holomorphic differential forms vanish. Our result can be seen as an explicit version of the Manin–Drinfeld theorem. Our method is to characterize such Eisenstein cycles as eigenvectors for the Hecke operators. We make crucial use of expressions of Hecke actions on modular symbols and on auxiliary level 2 structures.
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11F67 (primary)
title The Eisenstein cycles as modular symbols
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