Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral points on generalised affine Châtelet surfaces

Vladimir Mitankin

TL;DR

This work studies integral points on generalised affine Châtelet surfaces defined by $N_{\mathbb{Q}(\sqrt{a})/\mathbb{Q}}(x,y)=P(t)$, tying the existence of integral points to the Brauer–Manin obstruction. Assuming Schinzel's hypothesis, the authors show that, provided an archimedean unboundedness condition holds, the Brauer–Manin obstruction is the sole obstruction to the integral Hasse principle for these surfaces. The method combines a reduction to integral points on fibres via Conic bundles, a novel use of the narrow class group of $\mathbb{Q}(\sqrt{a})$, and Chebotarev density arguments to ensure the necessary local conditions. This extends classical results on Châtelet surfaces to a broader affine setting and highlights the role of arithmetic invariants like the narrow class group in integral points theory. Conditional on Hypothesis H, the paper also yields a strong local-global approximation property for adelic points with prescribed $t$-coordinates.

Abstract

We show, conditionally on Schinzel's hypothesis, that the only obstruction to the integral Hasse principle for generalised affine Châtelet surfaces is the Brauer--Manin one.

Integral points on generalised affine Châtelet surfaces

TL;DR

This work studies integral points on generalised affine Châtelet surfaces defined by , tying the existence of integral points to the Brauer–Manin obstruction. Assuming Schinzel's hypothesis, the authors show that, provided an archimedean unboundedness condition holds, the Brauer–Manin obstruction is the sole obstruction to the integral Hasse principle for these surfaces. The method combines a reduction to integral points on fibres via Conic bundles, a novel use of the narrow class group of , and Chebotarev density arguments to ensure the necessary local conditions. This extends classical results on Châtelet surfaces to a broader affine setting and highlights the role of arithmetic invariants like the narrow class group in integral points theory. Conditional on Hypothesis H, the paper also yields a strong local-global approximation property for adelic points with prescribed -coordinates.

Abstract

We show, conditionally on Schinzel's hypothesis, that the only obstruction to the integral Hasse principle for generalised affine Châtelet surfaces is the Brauer--Manin one.

Paper Structure

This paper contains 6 sections, 5 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. The Brauer--Manin obstruction
  4. Integral points on affine conics
  5. A Chebotarev density result
  6. Proof of Theorem \ref{['thm:Chatelet']}

Key Result

Theorem 1.1

Let $a \in \mathbb{Z}$ be non-zero squarefree and let $P_1(t), \dots, P_n(t) \in \mathbb{Z}[t]$ be irreducible such that $P(t) = \prod_{i = 1}^n P_i(t)$ is separable as a polynomial over $\mathbb{Q}$. Assume that $P_1(t)$ is linear or that the discriminant of the splitting field of $P_1(t)$ is copri Moreover, let $S$ be a finite set of finite places of $\mathbb{Q}$. Then for each $\varepsilon > 0$

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 1.2
  • Example 1.3
  • Example 1.4
  • Remark 3.1
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • ...and 3 more