The density of rational points on $\mathbb{P}^1$ with three stacky points

Abstract

In this paper we consider the density of rational points on the "stacky" curve X(P 1 ; 0, 2; 1, 2; ∞, 2) which is P 1 with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown height. In particular, we prove a conjecture of Ellenberg.

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

“…Beshaj, Gutierrez, and Shaska [8] have a definition of height on weighted projective space which agrees with ours in that case, as does the earlier preprint of Deng [21]. And in very recent work, Nasserden and Xiao [49] offer an alternative definition for stacky curves, and Ratko Darda [19] has proposed a definition for some other classes of stacks.…”

“…This set contains all pairs of coprime integers in [0, √ B], so it has size at least cB, as predicted. In fact, in recent work, Pierre Le Boudec (in personal communication) and Nasserden-Xiao [49] have independently shown that N U ,T X ,K (B) is bounded above and below by constant multiples of B log B 3 . This seems a very interesting case to explore further; can one obtain an asmyptotic N U ,T X ,K (B) ∼ cB log 3 B, and if so, what is the constant?…”

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

“…This set contains all pairs of coprime integers in [0, √ 𝐵], so it has size at least 𝑐𝐵, as predicted. In fact, in recent work, Pierre Le Boudec (in personal communication) and Nasserden-Xiao [53] have independently shown that 𝑁 U ,𝑇 X ,𝐾 (𝐵) is bounded above and below by constant multiples of 𝐵 log 𝐵 3 .…”