Skip to main content Accessibility help

Failures of weak approximation in families


Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.



Hide All
[BBD82] Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers , in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100 (Société Mathématique de France, Paris, 1982), 5171.
[Bha14] Bhargava, M., The geometric sieve and the density of squarefree values of invariant polynomials, Preprint (2014), arXiv:1402.0031.
[BCF13] Bhargava, M., Cremona, J. and Fisher, T., The proportion of plane cubic curves over $\mathbb{Q}$ that everywhere locally have a point, Int. J. Number Theory, to appear. Preprint (2013),arXiv:1311.5578.
[BSW15] Bhargava, M., Shankar, A. and Wang, J., Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces, Preprint (2015), arXiv:1512.03035.
[Bri02] Bright, M. J., Computations on diagonal quartic surfaces, PhD thesis, University of Cambridge (2002).
[Bri15] Bright, M. J., Bad reduction of the Brauer–Manin obstruction , J. Lond. Math. Soc. (2) 91 (2015), 643666.
[Bro01] Broberg, N., Rational points of cubic surfaces , in Rational points on algebraic varieties, Progress in Mathematics, vol. 199 (Birkhäuser, Basel, 2001), 1335.
[BH14] Browning, T. D. and Heath-Brown, D. R., Forms in many variables and differing degrees, J. Eur. Math. Soc. (JEMS), to appear. Preprint (2014), arXiv:1403.5937.
[Col03] Colliot-Thélène, J.-L., Points rationnels sur les fibrations , in Higher dimensional varieties and rational points (Budapest, 2001) (Springer, Berlin, 2003), 171221.
[CKS87] Colliot-Thélène, J.-L., Kanevsky, D. and Sansuc, J.-J., Arithmétique des surfaces cubiques diagonales , in Diophantine approximation and transcendence theory (Bonn, 1985), Lecture Notes in Mathematics, vol. 1290 (Springer, Berlin, 1987), 1108.
[CS77] Colliot-Thélène, J.-L. and Sansuc, J.-J., La R-équivalence sur les tores , Ann. Sci. Éc. Norm. Supér. 10 (1977), 175229.
[dlBBP12] de la Bretèche, R., Browning, T. D. and Peyre, E., On Manin’s conjecture for a family of Châtelet surfaces , Ann. of Math. (2) 175 (2012), 297343.
[Del80] Deligne, P., La conjecture de Weil, II , Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.
[DW13] Derenthal, U. and Wei, D., Strong approximation and descent, J. Reine Angew. Math., to appear. Preprint (2013), arXiv:1311.3914.
[Eke91] Ekedahl, T., An infinite version of the Chinese remainder theorem , Comment. Math. Univ. St. Pauli 40 (1991), 5359.
[FP13] Frei, C. and Pieropan, M., O-minimality on twisted universal torsors and Manin’s conjecture over number fields, Ann. Sci. Éc. Norm. Supér., to appear. Preprint (2013), arXiv:1312.6603.
[Gro68] Grothendieck, A., Le groupe de Brauer I, II, III , in Dix exposés sur la cohomologie des schémas (North-Holland, Amsterdam; Masson, Paris, 1968), 88188.
[Har94] Harari, D., Méthode des fibrations et obstruction de Manin , Duke Math. J. 75 (1994), 221260.
[Har97] Harari, D., Flèches de spécialisation en cohomologie étale et applications arithmétiques , Bull. Soc. Math. France 125 (1997), 143166.
[Har04] Harari, D., Weak approximation on algebraic varieties , in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progress in Mathematics, vol. 226 (Birkhäuser, Boston, 2004), 4360.
[JS15] Jahnel, J. and Schindler, D., On the Brauer–Manin obstruction for degree four del Pezzo surfaces, Preprint (2015), arXiv:1503.08292.
[Kle05] Kleiman, S., The Picard scheme , in Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, vol. 123 (American Mathematical Society, Providence, RI, 2005).
[LW54] Lang, S. and Weil, A., Number of points of varieties in finite fields , Amer. J. Math. 76 (1954), 819827.
[Lou13] Loughran, D., The number of varieties in a family which contain a rational point, J. Eur. Math. Soc. (JEMS), to appear. Preprint (2013), arXiv:1310.6219.
[Mar77] Marcus, D. A., Number fields (Springer, New York, 1977).
[Mil80] Milne, J. S., Étale cohomology (Princeton University Press, Princeton, NJ, 1980).
[Mil06] Milne, J. S., Arithmetic duality theorems, second edition (BookSurge, 2006).
[Neu99] Neukirch, J., Algebraic number theory (Springer, Berlin, 1999).
[PT01] Peyre, E. and Tschinkel, Y., Tamagawa numbers of diagonal cubic surfaces, numerical evidence , Math. Comp. 70 (2001), 367387.
[Pon95] Ponomaryov, K. N., Semialgebraic sets and variants of the Tarski–Seidenberg–Macintyre theorem , Algebra Logic 34 (1995), 182191.
[PS99] Poonen, B. and Stoll, M., The Cassels–Tate pairing on polarized abelian varieties , Ann. of Math. (2) 150 (1999), 11091149.
[PV04] Poonen, B. and Voloch, J. F., Random Diophantine equations , in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progress in Mathematics, vol. 226 (Birkhäuser, Boston, 2004), 175184.
[Rio14] Riou, J., Classes de Chern, morphismes de Gysin, pureté absolue, in Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Séminaire à l’École polytechnique 2006–2008), Astérisque, vols. 363–364 (Société Mathématique de France, 2014).
[Sch79] Schanuel, S., Heights in number fields , Bull. Soc. Math. France 107 (1979), 433449.
[Sch13] Schindler, D., Manin’s conjecture for certain biprojective hypersurfaces, J. Reine Angew. Math., to appear. Preprint (2013), arXiv:1307.7069.
[Ser90] Serre, J.-P., Spécialisation des éléments de Br2(Q(T 1, …, T n )) , C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 397402.
[Ser97] Serre, J.-P., Lectures on the Mordell–Weil theorem, third edition (F. Vieweg & Sohn, Braunschweig, 1997).
[SGA1] Grothendieck, A., Séminaire de géometrie algébrique du Bois-Marie SGA 1: revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224 (Springer, Berlin, 1961).
[SGA4½] Deligne, P., La classe de cohomologie associée à un cycle , in Séminaire de Géometrie Algébrique du Bois-Marie SGA 4½: Cohomologie étale, Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977).
[Sko96] Skorobogatov, A., Descent on fibrations over the projective line , Amer. J. Math. 118 (1996), 905923.
[Sko01] Skorobogatov, A., Torsors and rational points (Cambridge University Press, Cambridge, 2001).
[Spa95] Spain, P. G., Lipschitz2 : a new version of an old principle , Bull. Lond. Math. Soc. 27 (1995), 565566.
[Sta14] The Stacks Project Authors, Stacks Project (2014),
[Uem14] Uematsu, T., On the Brauer group of diagonal cubic surfaces , Quart. J. Math. 65 (2014), 677701.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO

Failures of weak approximation in families


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.