Serre's problem on the density of isotropic fibres in conic bundles
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)].NEWLINENEWLINEThe divisor sums encountered take the shape NEWLINE\[NEWLINE\sum_{|s|,|t|\leq B}\;\prod_{i=1}^n\;\sum_{d_i|\Delta_i(s,t)} \left(\frac{F_i(s,t)}{d_i}\right)NEWLINE\]NEWLINE where the Jacobi symbols involve forms \(F_i\) of even degree, and where the \(\Delta_i\) are forms whose degrees total at most 3. There have been many investigations into special cases of such sums. However even for the case in which \(n=1\) and \(F_1=1\) one currently achieves a power saving in \(B\) only when \(\Delta_1\) has degree at most 3. Since a power saving is required for the method in the present paper, this explains the limitation on the rank of \(\pi\).
scientific article; zbMATH DE number 6642540
scientific article