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Failure of the Hasse principle for Enriques surfaces. (English) Zbl 1239.11068
Following Manin’s original work describing how the Brauer group of a variety can obstruct the existence of rational points, his construction (known as the Brauer–Manin obstruction) has been generalised in various ways to give several different obstructions. Those considered in this article, in increasing order of strength, are: the algebraic Brauer–Manin obstruction; the Brauer–Manin obstruction; and the Brauer–Manin obstruction applied to finite étale torsors, known as the étale-Brauer obstruction. Various authors have constructed examples in which these obstructions give genuinely different information about the rational points.
In this well-explained article, the authors fill a gap in the literature by constructing an Enriques surface \(X\) over the rational numbers on which the étale-Brauer obstruction shows that there are no rational points, yet there is no algebraic Brauer–Manin obstruction to the Hasse principle. This leaves open the possibility that the absence of rational points is explained by the (necessarily transcendental) Brauer–Manin obstruction; the authors relate this to the corresponding question for a \(K3\) surface. In the notation of the article:
Theorem 1.1. There exists an Enriques surface \(X/\mathbb Q\) such that \[ X(\mathbb A_\mathbb Q)^{\text{ét},\mathbf{Br}} = \emptyset \qquad \text{and} \qquad X(\mathbb A_\mathbb Q)^{\mathbf{Br}_1} \neq \emptyset. \] Moreover, if \(X(\mathbb A_{\mathbb Q})^{\mathbf{Br}} = \emptyset\), then \(Y(\mathbb A_\mathbb Q)^{\mathbf{Br}\, Y \setminus \mathbf{Br}_1 Y} = \emptyset\), where \(Y\) is a \(K3\) double cover of \(X\).
The variety \(X\) in question is defined as the quotient of a \(K3\) surface \(Y\) by a fixed-point free involution; \(Y\) itself is explicitly given as the intersection of three quadrics in \({\mathbb P}^5\), parametrised by three positive integers \(a,b,c\) subject to certain arithmetic conditions (described in Theorem 1.2). In particular, \((a,b,c)=(12,111,13)\) satisfies the conditions.
Since \(Y\) is simply connected, the only finite étale torsors over \(X\) are twists of \(Y \to X\). To show that there is an étale-Brauer obstruction to the existence of rational points on \(X\), it is enough to show that there is a Brauer–Manin obstruction to the existence of rational points on every twist of \(Y\); the construction of \(Y\) ensures that this comes down to a known example of a Brauer–Manin obstruction, given by B. J. Birch and H. P. F. Swinnerton-Dyer [J. Reine Angew. Math. 274–275, 164–174 (1975; Zbl 0326.14007)].
Proving the absence of an algebraic Brauer–Manin obstruction to rational points on \(X\) is more intricate, and involves computing \(\mathbf{Pic} (\bar{X})\) and its Galois cohomology. This is accomplished using a fibration of \(Y\) into genus 1 curves; explicit equations for divisors, and the Galois action on them, are given in appendices.

11G35 Varieties over global fields
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14F22 Brauer groups of schemes
14J28 \(K3\) surfaces and Enriques surfaces
Full Text: DOI arXiv
[1] Artin, M.; Swinnerton-Dyer, H.P.F., The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. math., 20, 249-266, (1973), MR0417182 (54 #5240) · Zbl 0289.14003
[2] Basile, Carmen Laura; Skorobogatov, Alexei N., On the Hasse principle for bielliptic surfaces, (), 31-40, MR2053453 (2005e:14056) · Zbl 1073.14052
[3] Beauville, Arnaud, Complex algebraic surfaces, London math. soc. stud. texts, vol. 34, (1996), Cambridge Univ. Press Cambridge, translated from the 1978 French original by R. Barlow, with assistance from N.I. Shepherd-Barron and M. Reid, MR1406314 (97e:14045) · Zbl 0849.14014
[4] Beauville, Arnaud, On the Brauer group of Enriques surfaces, Math. res. lett., 16, 6, 927-934, (2009), MR2576681 · Zbl 1195.14053
[5] Birch, B.J.; Swinnerton-Dyer, H.P.F., The Hasse problem for rational surfaces, J. reine angew. math., 274/275, 164-174, (1975), collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III, MR0429913 (55 #2922) · Zbl 0326.14007
[6] Bosma, Wieb; Cannon, John; Playoust, Catherine, The magma algebra system. I. the user language, Computational algebra and number theory, London, 1993, J. symbolic comput., 24, 3-4, 235-265, (1997), MR1484478 · Zbl 0898.68039
[7] Colliot-Thélène, J.-L.; Sansuc, J.-J., La descente sur LES variétés rationnelles, (), 223-237, (in French), MR605344 (82d:14016) · Zbl 0451.14018
[8] Cossec, François R., Projective models of Enriques surfaces, Math. ann., 265, 3, 283-334, (1983), MR721398 (86d:14035) · Zbl 0501.14021
[9] Cossec, François R.; Dolgachev, Igor V., Enriques surfaces. I, Progr. math., vol. 76, (1989), Birkhäuser Boston Inc. Boston, MA, MR986969 (90h:14052) · Zbl 0665.14017
[10] Steven Cunnane, Rational points on Enriques surfaces, Imperial College London, PhD thesis, 2007.
[11] Elsenhans, Andreas-Stephan; Jahnel, Jörg, On the Weil polynomials of K3 surfaces, (), 126-141 · Zbl 1260.11046
[12] Flynn, E.V., The Hasse principle and the Brauer-Manin obstruction for curves, Manuscripta math., 115, 4, 437-466, (2004), MR2103661 (2005j:11047) · Zbl 1069.11023
[13] Harari, D., Obstructions de Brauer-Manin transcendantes, (), 75-87 · Zbl 0926.14009
[14] Harari, David; Skorobogatov, Alexei N., Non-abelian descent and the arithmetic of Enriques surfaces, Int. math. res. not. IMRN, 52, 3203-3228, (2005), MR2186792 (2006m:14031) · Zbl 1099.14008
[15] Hassett, Brendan; Várilly-Alvarado, Anthony; Varilly, Patrick, Transcendental obstructions to weak approximation on general K3 surfaces, (2010), preprint · Zbl 1228.14030
[16] Ieronymou, Evis, Diagonal quartic surfaces and transcendental elements of the Brauer group, (2009), preprint · Zbl 1263.14023
[17] Kresch, Andrew; Tschinkel, Yuri, Effectivity of Brauer-Manin obstructions on surfaces, (2010), preprint · Zbl 1236.14027
[18] Carl-Erik Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, thesis, University of Uppsala, 1940, 97 (in German), MR0022563 (9,225c). · Zbl 0025.24802
[19] Liu, Qing; Lorenzini, Dino; Raynaud, Michel, On the Brauer group of a surface, Invent. math., 159, 3, 673-676, (2005), MR2125738 (2005k:14036) · Zbl 1077.14023
[20] Manin, Y.I., Le groupe de Brauer-Grothendieck en géométrie diophantienne, (), 401-411, MR0427322 (55 #356)
[21] Milne, J.S., On a conjecture of Artin and Tate, Ann. of math. (2), 102, 3, 517-533, (1975), MR0414558 (54 #2659) · Zbl 0343.14005
[22] Poonen, Bjorn, Heuristics for the Brauer-Manin obstruction for curves, Experiment. math., 15, 4, 415-420, (2006), MR2293593 (2008d:11062) · Zbl 1173.11040
[23] Poonen, Bjorn, Insufficiency of the Brauer-Manin obstruction applied to étale covers, Ann. of math., 171, 3, 2157-2169, (2010) · Zbl 1284.11096
[24] Reichardt, Hans, Einige im kleinen überall lösbare, im großen unlösbare diophantische gleichungen, J. reine angew. math., 184, 12-18, (1942), (in German), MR0009381 (5,141c) · Zbl 0026.29701
[25] Sarnak, Peter; Wang, Lan, Some hypersurfaces in \(\mathbf{P}^4\) and the Hasse-principle, C. R. acad. sci. Paris Sér. I math., 321, 3, 319-322, (1995), (in English, with English and French summaries), MR1346134 (96j:14014) · Zbl 0857.14013
[26] Victor Scharaschkin, Local global problems and the Brauer-Manin obstruction, University of Michigan, PhD thesis, 1999.
[27] Serre, Jean-Pierre, Linear representations of finite groups, Grad. texts in math., vol. 42, (1977), Springer-Verlag New York, translated from the second French edition by Leonard L. Scott, MR0450380 (56 #8675)
[28] Skorobogatov, Alexei N., Beyond the Manin obstruction, Invent. math., 135, 2, 399-424, (1999), MR1666779 (2000c:14022) · Zbl 0951.14013
[29] Skorobogatov, Alexei N., Torsors and rational points, Cambridge tracts in math., vol. 144, (2001), Cambridge Univ. Press Cambridge, MR1845760 (2002d:14032) · Zbl 0972.14015
[30] Skorobogatov, Alexei; Swinnerton-Dyer, Peter, 2-descent on elliptic curves and rational points on certain Kummer surfaces, Adv. math., 198, 2, 448-483, (2005), MR2183385 (2006g:11129) · Zbl 1085.14021
[31] Stoll, Michael, Finite descent obstructions and rational points on curves, Algebra number theory, 1, 4, 349-391, (2007), MR2368954 (2008i:11086) · Zbl 1167.11024
[32] Stoll, Michael; Testa, Damiano, The surface parametrizing cuboids, (2010), preprint
[33] van Luijk, Ronald, K3 surfaces with Picard number one and infinitely many rational points, Algebra number theory, 1, 1, 1-15, (2007), MR2322921 (2008d:14058) · Zbl 1123.14022
[34] Verra, Alessandro, The étale double covering of an Enriques surface, Rend. semin. mat. univ. politec. Torino, 41, 3, 131-167, (1983), (1984), MR778864 (86m:14029) · Zbl 0581.14027
[35] Wittenberg, Olivier, Transcendental Brauer-Manin obstruction on a pencil of elliptic curves, (), 259-267, MR2029873 (2005c:11082) · Zbl 1173.11336
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