The density of fibres with a rational point for a fibration over hypersurfaces of low degree
[La densité des fibres possédant un point rationnel pour une fibration au-dessus d’une hypersurface de bas degré]
Annales de l'Institut Fourier, Online first, 31 p.

Nous établissons une formule asymptotique concernant la proportion de fibres possédant un point rationnel dans le cas d’une fibration en coniques, la base de la fibration étant une hypersurface générique de bas degré.

We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The base of the fibration is a general hypersurface of low degree.

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DOI : https://doi.org/10.5802/aif.3413
Classification : 14G05,  14D06,  11P55,  14D10
Mots clés : Méthode du cercle de Hardy-Littlewood, Le problème de Serre, Fibres possédant un point rationnel
@unpublished{AIF_0__0_0_A6_0,
     author = {Sofos, Efthymios and Visse-Martindale, Erik},
     title = {The density of fibres with a rational point for a fibration over hypersurfaces of low degree},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3413},
     language = {en},
     note = {Online first},
}
Sofos, Efthymios; Visse-Martindale, Erik. The density of fibres with a rational point for a fibration over hypersurfaces of low degree. Annales de l'Institut Fourier, Online first, 31 p.

[1] Birch, Bryan J. Forms in many variables, Proc. R. Soc. Lond., Ser. A, Volume 265 (1962), pp. 245-263 | Article | MR 150129 | Zbl 0103.03102

[2] Browning, Tim; Loughran, Daniel Sieving rational points on varieties, Trans. Am. Math. Soc., Volume 371 (2019) no. 8, pp. 5757-5785 | Article | MR 3937309 | Zbl 1412.14015

[3] Frei, Christopher; Loughran, Daniel; Sofos, Efthymios Rational points of bounded height on general conic bundle surfaces, Proc. Lond. Math. Soc., Volume 117 (2018) no. 2, pp. 407-440 | Article | MR 3851328 | Zbl 06929624

[4] Friedlander, John; Iwaniec, Henryk Opera de cribro, Colloquium Publications, 57, American Mathematical Society, 2010, xx+527 pages | Article | MR 2647984 | Zbl 1226.11099

[5] Hooley, Christopher On ternary quadratic forms that represent zero, Glasg. Math. J., Volume 35 (1993) no. 1, pp. 13-23 | Article | MR 1199934 | Zbl 0774.11019

[6] Hooley, Christopher On ternary quadratic forms that represent zero. II, J. Reine Angew. Math., Volume 602 (2007), pp. 179-225 | Article | MR 2300456 | Zbl 1146.11023

[7] Loughran, Daniel The number of varieties in a family which contain a rational point, J. Eur. Math. Soc., Volume 20 (2018) no. 10, pp. 2539-2588 | Article | MR 3852186 | Zbl 1452.14018

[8] Loughran, Daniel; Smeets, Arne Fibrations with few rational points, Geom. Funct. Anal., Volume 26 (2016) no. 5, pp. 1449-1482 | Article | MR 3568035 | Zbl 1357.14028

[9] Loughran, Daniel; Takloo-Bighash, Ramin; Tanimoto, Sho Zero-loci of Brauer group elements on semi-simple algebraic groups, J. Inst. Math. Jussieu, Volume 19 (2020) no. 5, pp. 1467-1507 | Article | MR 4138949 | Zbl 1454.14063

[10] Peyre, Emmanuel; Tschinkel, Yuri Tamagawa numbers of diagonal cubic surfaces, numerical evidence, Math. Comput., Volume 70 (2001) no. 233, pp. 367-387 | Article | MR 1681100 | Zbl 0961.14012

[11] Poonen, Bjorn; Voloch, José Felipe Random Diophantine equations, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002) (Progress in Mathematics), Volume 226, Birkhäuser, 2004, pp. 175-184 (With appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz) | Article | MR 2029869 | Zbl 1208.11050

[12] Rieger, Georg J. Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. II, J. Reine Angew. Math., Volume 217 (1965), pp. 200-216 | Article | MR 174533 | Zbl 0141.04305

[13] Schmidt, Wolfgang M. Simultaneous rational zeros of quadratic forms, Seminar on Number Theory (Paris, 1980/1981) (Progress in Mathematics), Volume 22, Birkhäuser, 1982, pp. 281-307 | MR 693325 | Zbl 0492.10017

[14] Serre, Jean-Pierre A course in arithmetic, Graduate Texts in Mathematics, 7, Springer, 1973, viii+115 pages (Translated from the French) | MR 0344216 | Zbl 0256.12001

[15] Serre, Jean-Pierre Spécialisation des éléments de Br 2 (Q(T 1 ,,T n )), C. R. Math. Acad. Sci. Paris, Volume 311 (1990) no. 7, pp. 397-402 | MR 1075658

[16] Skinner, Christopher M. Forms over number fields and weak approximation, Compos. Math., Volume 106 (1997) no. 1, pp. 11-29 | Article | MR 1446148 | Zbl 0892.11014

[17] Sofos, Efthymios Serre’s problem on the density of isotropic fibres in conic bundles, Proc. Lond. Math. Soc., Volume 113 (2016) no. 2, pp. 261-288 | Article | MR 3534973 | Zbl 1355.14019

[18] Tenenbaum, Gérald Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, 163, American Mathematical Society, 2015, xxiv+629 pages (Translated from the 2008 French edition by Patrick D. F. Ion) | MR 3363366 | Zbl 1336.11001

[19] Visse-Martindale, Erik Counting points on K3 surfaces and other arithmetic-geometric objects (2019) (https://openaccess.leidenuniv.nl/handle/1887/67532) (Ph. D. Thesis)