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Generic diagonal conic bundles revisited

Alexei N. Skorobogatov, Efthymios Sofos

Abstract

We prove a stronger form of our previous result that Schinzel's Hypothesis holds for $100\%$ of $n$-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.

Generic diagonal conic bundles revisited

Abstract

We prove a stronger form of our previous result that Schinzel's Hypothesis holds for of -tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.
Paper Structure (3 sections, 11 theorems, 72 equations)

This paper contains 3 sections, 11 theorems, 72 equations.

Table of Contents

  1. Schinzel hypothesis on average with quadratic residue conditions
  2. Brauer group of generic diagonal conic bundles
  3. Explicit densities

Key Result

Theorem 1.1

Fix any $(d_1,\ldots,d_n) \in \mathbb N^n$, $\varepsilon>0$, $M\in \mathbb N$, $m_0\in {\mathbb Z}/M$ and ${\bf Q}\in (({\mathbb Z}/M)[t])^n$ such that ${\rm{deg}}(Q_i)\leq d_i$ and $\gcd(Q_i(m_0), M)=1$ for all $i=1,\ldots, n$. For every $(i,j)\in (\mathbb N \cap[1,n])^2$ with $i<j$ let $\epsilon

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 1.2: Cassels
  • Corollary 1.3
  • Lemma 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 3.1
  • Remark 3.2
  • ...and 2 more