Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.20316

S-squareful values of binary quadratic forms

Sarah Sophie Pohl
View original source ↗Read source document ↗Export metadata ↓

Abstract

Shute showed that the leading constant in the asymptotic formula for squareful values of the form $ax^2+by^2$ does not match with the conjecture established by Pieropan, Smeets, Tanimoto and Várilly-Alvarado. In this paper, $S$-squareful values of this form are considered for $S$ the set of primes that divide $a$ and $b$. We prove an asymptotic formula for these numbers and show that the leading constant of the asymptotic formula still does not match the conjecture. Furthermore, we consider the updated conjecture established by Chow, Loughran, Takloo-Bighash, and Tanimoto regarding the leading constant for Campana points in the given setting and show that this constant is consistent with our computed constant.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sarah Sophie Pohl. 2026-10-14. S-squareful values of binary quadratic forms. https://arxiv.org/abs/2610.20316

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes

Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip.

math.NT↗
arXiv2026

Root numbers for twisted Fermat quotient curves II

This is a sequel to the previous work of the author Yanagihara (2025). Let $\ell$ be an odd prime, let $N \geq 1$ be an integer, and let $δ\geq 1$ be an $\ell^N$-th-power-free integer. Let $r,s,t>0$ be integers satisfying $r+s+t=\ell^N$. In Yanagihara (2025), the author computed the root number of the Fermat quotient curve $y^{\ell^N}=x^r(δ-x)^s$ under the assumptions that $\ell\nmid rst$ and that $\operatorname{ord}_{\ell}(δ)=0$ or $\ell\nmid \operatorname{ord}_{\ell}(δ)$. In this paper, we study the case where the technical assumption $\ell\nmid rst$ is dropped. As one such case, we compute the root number when $\ell^{N-1}\| r$ and $\ell\nmid stδ$.

math.NT↗
arXiv2026

Potential semistability of Finite height Galois representations: Relative case

Let $K$ be a $p$-adic field. We define the notion of finite height for an étale $\mathbb{Z}_p$-local system on a smooth affinoid adic space $\mathcal{X}$ over $K$ with semistable reduction. Using analytic prismatic $F$-crystals and purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603), we prove that if an étale $\mathbb{Z}_p$-local system over $\mathcal{X}$ is of finite height then its pullback along a finite étale cover of $\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.

math.NT↗