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arXiv · 2108.04411

Heights and quantitative arithmetic on stacky curves

Brett Nasserden·Stanley Yao Xiao
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Abstract

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space $\mathbb{P}^1$, is equivalent to the $abc$-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

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Brett Nasserden, Stanley Yao Xiao. 2021-08-10. Heights and quantitative arithmetic on stacky curves. https://doi.org/10.4153/s0008414x24000075

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