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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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CiteScore 2018: 0.71

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1435-5337
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Volume 31, Issue 3

# Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

Fu-Tsun Wei
/ Takao Yamazaki
Published Online: 2019-01-20 | DOI: https://doi.org/10.1515/forum-2018-0141

## Abstract

We consider the generalized Jacobian $\stackrel{~}{J}$ of the modular curve ${X}_{0}\left(N\right)$ of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the $ℚ$-rational torsion points on $\stackrel{~}{J}$ up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on $\stackrel{~}{J}$ and its Eisenstein property.

MSC 2010: 11G09; 11G18; 11F03; 14H40; 14G35

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Revised: 2018-11-28

Published Online: 2019-01-20

Published in Print: 2019-05-01

Funding Source: Ministry of Science and Technology, Taiwan

Award identifier / Grant number: 105-2115-M-007-018-MY2

Award identifier / Grant number: 107-2628-M-007-004-MY4

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15K04773

The first author is supported by Ministry of Science and Technology, Taiwan (grant number 105-2115-M-007-018-MY2 and 107-2628-M-007-004-MY4). The second author is supported by Japan Society for the Promotion of Science KAKENHI Grant (grant number 15K04773).

Citation Information: Forum Mathematicum, Volume 31, Issue 3, Pages 647–659, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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