Serre's problem on the density of isotropic fibres in conic bundles
type: Item
instance of: scholarly article
title: Serre's problem on the density of isotropic fibres in conic bundles
zbMATH Open document ID: 1355.14019
author: Efthymios Sofos
publication date: 2016-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1602.03140
full work available at URL: 198842.pdf
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 14D10
zbMATH DE Number: 6642540
zbMATH Keywords: conic bundle
zbMATH Keywords: isotropic fibre
zbMATH Keywords: Serre problem
zbMATH Keywords: divisor sum
zbMATH Keywords: Hooley neutralizer
zbMATH Keywords: lower bound
zbMATH Keywords: sieve
MaRDI profile type: Publication
OpenAlex ID: W2260958437
arXiv ID: 1602.03140
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cites work: Q3705555
cites work: Q3423329
cites work: Q4668450
cites work: Descent on fibrations over the projective line
DOI: 10.1112/PLMS/PDW030
published in: Proceedings of the London Mathematical Society. Third Series
review text: Let \(\pi: X\rightarrow\mathbb{P}^1_{\mathbb{Q}}\) be a nonsingular conic bundle, with \(n\) non-split fibres. Let NEWLINE\[NEWLINEN(B):=\#\{x\in\mathbb{P}^1_{\mathbb{Q}}: H(x)\leq B,\,\pi^{-1}(x) \text{ has a rational point}\},NEWLINE\]NEWLINE where \(H(x)\) is the usual Weil height. For example, one might ask, for how many coprime pairs \(s,t\) of integers with \(\max(|s|,|t|)\leq B\) does the equation NEWLINE\[NEWLINEsY_1^2+t(Y_2^2+Y_3^2)=0NEWLINE\]NEWLINE have a non-trivial solution? Here \((s,t)=(1,0)\) gives a split fibre, and \((s,t)=(0,1)\) produces a non-split one, so that \(n=1\).NEWLINENEWLINE\textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 311, No. 7, 397--402 (1990; Zbl 0711.13002)] used a direct application of the large sieve to show that NEWLINE\[NEWLINEN(B)\ll B^2(\log B)^{-n/2},NEWLINE\]NEWLINE and he asked whether the corresponding lower bound also holds. Here one should require that there is at least one smooth fibre which is isotropic over \(\mathbb{Q}\), as is clearly necessary.NEWLINENEWLINEThe main result of the paper is that NEWLINE\[NEWLINEN(B)\asymp \frac{B^2}{(\log B)^{n/2}}NEWLINE\]NEWLINE under the above assumption, whenever the ``rank'' of \(\pi\) is at most 3. Here the rank is defined as the sum of the degrees of the field extensions corresponding to the non-split fibres, following \textit{A. N. Skorobogatov} [Am. J. Math. 118, No. 5, 905--923 (1996; Zbl 0880.14008)]. In the example above the rank is \([\mathbb{Q}(i):\mathbb{Q}]=2\).NEWLINENEWLINEFor the proof the author first constructs arithmetic expressions for the characteristic function for isotropic fibres, using the Hasse principle for conics. Summing these up produces divisor sums weighted by awkward factors of the shape \(2^{-\omega(k)}\). To handle these one uses a lower bound constructed from bound sieve weights. This idea, referred to as ``Hooley's neutralizer'' is taken from \textit{C. Hooley}'s work [J. Reine Angew. Math. 267, 207--218 (1974; Zbl 0283.10031)].NEWLINE…
reviewed by: D. R. Heath-Brown
description: scientific article; zbMATH DE number 6642540