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Local solubility in generalised Châtelet varieties

Kevin Destagnol, Julian Lyczak, Efthymios Sofos

TL;DR

This work develops a multivariate circle-method framework to obtain asymptotics for averages of arithmetic functions evaluated at polynomial values in several variables, and applies it to count rational fibers in families of high-dimensional Châtelet varieties with potentially large subordinate Brauer groups. The authors introduce a splitting trick and a robust equidistribution analysis to express the leading constant as a finite sum of Euler products, linking it to Tamagawa measures and subordinate Brauer data. They verify convergence of the leading constant and compute the Tamagawa-related factors, ultimately matching the asymptotic with Peyre-type predictions for the adelic volume and the Brauer-manin obstruction. The results advance arithmetic statistics for multivariate norm-type problems, showing how ramified Brauer phenomena and Tamagawa densities interact in high-dimensional fibrations and providing new avenues for Chowla/Bateman–Horn-type applications in several variables.

Abstract

We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the number of fibers with a rational point in families of high-dimensional Châtelet varieties, allowing for arbitrarily large subordinate Brauer groups.

Local solubility in generalised Châtelet varieties

TL;DR

This work develops a multivariate circle-method framework to obtain asymptotics for averages of arithmetic functions evaluated at polynomial values in several variables, and applies it to count rational fibers in families of high-dimensional Châtelet varieties with potentially large subordinate Brauer groups. The authors introduce a splitting trick and a robust equidistribution analysis to express the leading constant as a finite sum of Euler products, linking it to Tamagawa measures and subordinate Brauer data. They verify convergence of the leading constant and compute the Tamagawa-related factors, ultimately matching the asymptotic with Peyre-type predictions for the adelic volume and the Brauer-manin obstruction. The results advance arithmetic statistics for multivariate norm-type problems, showing how ramified Brauer phenomena and Tamagawa densities interact in high-dimensional fibrations and providing new avenues for Chowla/Bateman–Horn-type applications in several variables.

Abstract

We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the number of fibers with a rational point in families of high-dimensional Châtelet varieties, allowing for arbitrarily large subordinate Brauer groups.

Paper Structure

This paper contains 20 sections, 32 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Averages of arithmetic functions over values of multivariable polynomials
  3. Proving Theorem \ref{['thm:mainthrm']} via a Splitting Trick
  4. Previous results
  5. The circle method tool
  6. Averages of arithmetic functions over values of multivariable polynomials
  7. A uniform upper-bound
  8. Proof of Theorem \ref{['lem:vachms']}
  9. Equidistribution and other analytic lemmas
  10. Equidistribution
  11. Analytic lemmas
  12. Proof of the asymptotic
  13. Preparing the grounds
  14. Application of the circle method tool
  15. Convergence of the leading constant
  16. ...and 5 more sections

Key Result

Theorem 1.1

Fix any $A>0$. Then for $B\geqslant 3$ we have where $\zeta$ is the Riemann zeta function and $\gamma$ is defined by To describe $\gamma(\mathbf s )$ we let $\mu_p$ be the $p$-adic Haar measure on $\mathbb{Z}_p^{n+1}$ for a prime $p$ and let $\prod_{p\leqslant T} \mu_p$ be the product measure on $\prod_{p\leqslant T} \mathbb{Z}_p^{n+1}$. Then where $(\cdot,\cdot)_{\mathbb{Q}_p}$ is the Hilbert

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.2: Leading constant
  • Remark 1.3: Brauer group
  • Remark 1.4: Reciprocity at infinitely many primes
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5: Applications
  • Remark 2.6
  • ...and 59 more