Heights and quantitative arithmetic on stacky curves
Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 481-534
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In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$-conjecture.
Nasserden, Brett; Xiao, Stanley Yao. Heights and quantitative arithmetic on stacky curves. Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 481-534. doi: 10.4153/S0008414X24000075
@article{10_4153_S0008414X24000075,
author = {Nasserden, Brett and Xiao, Stanley Yao},
title = {Heights and quantitative arithmetic on stacky curves},
journal = {Canadian journal of mathematics},
pages = {481--534},
year = {2025},
volume = {77},
number = {2},
doi = {10.4153/S0008414X24000075},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000075/}
}
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%0 Journal Article %A Nasserden, Brett %A Xiao, Stanley Yao %T Heights and quantitative arithmetic on stacky curves %J Canadian journal of mathematics %D 2025 %P 481-534 %V 77 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000075/ %R 10.4153/S0008414X24000075 %F 10_4153_S0008414X24000075
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