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The Eisenstein cycles as modular symbols
Author(s) -
Banerjee Debargha,
Merel Loïc
Publication year - 2018
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12136
Subject(s) - mathematics , eisenstein series , modular form , hecke operator , holomorphic function , modular design , pure mathematics , integer (computer science) , eigenvalues and eigenvectors , modular group , algebra over a field , arithmetic , computer science , physics , quantum mechanics , programming language , operating system
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.