REFINED MODULAR SYMBOLS AND REFINED EISENSTEIN IDEALS
Abstract
We introduce refinements of the theory of modular symbols for a subgroup Γ of SL 2 (Z). The group H
] by the Manin relations. Our more general refinement consists in replacing the Manin relations by what we call the Lewis relation. We apply this first to Γ = Γ 0 (N ), for N prime. The Hecke algebra acting on refined modular symbols has an Eisenstein ideal I L , which refines Mazur's classical situation. If N ̸ ≡ 1 (mod 8), we have
The second application concerns Γ = Γ 1 (N ), for any integer N . Sharifi has constructed a map Π
, where ζ N is a primitive N -th root of unity. Sharifi and Venkatesh have established that such a map is Eisenstein. Relying on their work, we construct a lift of Π • N on refined modular symbols to K 2 (Z[ζ N , 1/N ]) that retains the Eisenstein property.
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BibTeX
@misc{lecouturier2026refined,
title = {{REFINED MODULAR SYMBOLS AND REFINED EISENSTEIN IDEALS}},
author = {Emmanuel Lecouturier and Loïc Merel},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05752071v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - REFINED MODULAR SYMBOLS AND REFINED EISENSTEIN IDEALS
AU - Emmanuel Lecouturier
AU - Loïc Merel
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05752071v1
AB - We introduce refinements of the theory of modular symbols for a subgroup Γ of SL 2 (Z). The group H ] by the Manin relations. Our more general refinement consists in replacing the Manin relations by what we call the Lewis relation. We apply this first to Γ = Γ 0 (N ), for N prime. The Hecke algebra acting on refined modular symbols has an Eisenstein ideal I L , which refines Mazur's classical situation. If N ̸ ≡ 1 (mod 8), we have The second application concerns Γ = Γ 1 (N ), for any integer N . Sharifi has constructed a map Π , where ζ N is a primitive N -th root of unity. Sharifi and Venkatesh have established that such a map is Eisenstein. Relying on their work, we construct a lift of Π • N on refined modular symbols to K 2 (Z[ζ N , 1/N ]) that retains the Eisenstein property.
ER -