Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 
 
 

 

A positive proportion of locally soluble hyperelliptic curves over $ \mathbb{Q}$ have no point over any odd degree extension


Authors: Manjul Bhargava, Benedict H. Gross and Xiaoheng Wang; with an appendix by Tim Dokchitser; Vladimir Dokchitser
Journal: J. Amer. Math. Soc. 30 (2017), 451-493
MSC (2000): Primary 11G30; Secondary 14G05
DOI: https://doi.org/10.1090/jams/863
Published electronically: July 27, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A hyperelliptic curve over $ \mathbb{Q}$ is called ``locally soluble'' if it has a point over every completion of $ \mathbb{Q}$. In this paper, we prove that a positive proportion of hyperelliptic curves over $ \mathbb{Q}$ of genus $ g\geq 1$ are locally soluble but have no points over any odd degree extension of $ \mathbb{Q}$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $ k > 0$, the proportion of locally soluble hyperelliptic curves over $ \mathbb{Q}$ of genus $ g$ having no points over any odd degree extension of $ \mathbb{Q}$ of degree at most $ k$ tends to $ 1$ as $ g$ tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to $ 2$, together with suitable arguments from the geometry of numbers.


References [Enhancements On Off] (What's this?)

  • [1] M. Bhargava, Most hyperelliptic curves over $ {\mathbb{Q}}$ have no rational points, http://arxiv.org/abs/1308.0395v1.
  • [2] M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, http://arxiv.org/abs/1402.0031v1.
  • [3] M. Bhargava, J. Cremona, and T. Fisher, The density of hyperelliptic curves over $ {\mathbb{Q}}$ of genus $ g$ that have points everywhere locally, preprint.
  • [4] Manjul Bhargava and Benedict H. Gross, Arithmetic invariant theory, Symmetry: representation theory and its applications, Progr. Math., vol. 257, Birkhäuser/Springer, New York, 2014, pp. 33-54. MR 3363006, https://doi.org/10.1007/978-1-4939-1590-3_3
  • [5] Manjul Bhargava and Benedict H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, Automorphic representations and $ L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 23-91. MR 3156850
  • [6] M. Bhargava, B. Gross, and X. Wang, Arithmetic invariant theory II, Progress in Mathematics, Representations of Lie Groups: In Honor of David A Vogan, Jr. on his 60th Birthday, to appear.
  • [7] Manjul Bhargava and Arul Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181 (2015), no. 1, 191-242. MR 3272925, https://doi.org/10.4007/annals.2015.181.1.3
  • [8] Manjul Bhargava and Arul Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2) 181 (2015), no. 2, 587-621. MR 3275847, https://doi.org/10.4007/annals.2015.181.2.4
  • [9] B. J. Birch and J. R. Merriman, Finiteness theorems for binary forms with given discriminant, Proc. Lond. Math. Soc. (3) 24 (1972), 385-394. MR 0306119
  • [10] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Berlin, New York: Springer-Verlag, 1990.
  • [11] Nils Bruin and Michael Stoll, Two-cover descent on hyperelliptic curves, Math. Comp. 78 (2009), no. 268, 2347-2370. MR 2521292, https://doi.org/10.1090/S0025-5718-09-02255-8
  • [12] J. W. S. Cassels, The Mordell-Weil group of curves of genus $ 2$, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser, Boston, Mass., 1983, pp. 27-60. MR 717589
  • [13] Jean-Louis Colliot-Thélène and Bjorn Poonen, Algebraic families of nonzero elements of Shafarevich-Tate groups, J. Amer. Math. Soc. 13 (2000), no. 1, 83-99. MR 1697093, https://doi.org/10.1090/S0894-0347-99-00315-X
  • [14] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375-492 (French). MR 899402, https://doi.org/10.1215/S0012-7094-87-05420-2
  • [15] Amir Dembo, Bjorn Poonen, Qi-Man Shao, and Ofer Zeitouni, Random polynomials having few or no real zeros, J. Amer. Math. Soc. 15 (2002), no. 4, 857-892 (electronic). MR 1915821, https://doi.org/10.1090/S0894-0347-02-00386-7
  • [16] U. V. Desale and S. Ramanan, Classification of vector bundles of rank $ 2$ on hyperelliptic curves, Invent. Math. 38 (1976/77), no. 2, 161-185. MR 0429897
  • [17] Tim Dokchitser and Vladimir Dokchitser, Self-duality of Selmer groups, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 257-267. MR 2475965, https://doi.org/10.1017/S0305004108001989
  • [18] Tim Dokchitser and Vladimir Dokchitser, Regulator constants and the parity conjecture, Invent. Math. 178 (2009), no. 1, 23-71. MR 2534092, https://doi.org/10.1007/s00222-009-0193-7
  • [19] Ron Donagi, Group law on the intersection of two quadrics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (1980), no. 2, 217-239. MR 581142
  • [20] N. N. Dong Quan, Algebraic families of hyperelliptic curves violating the Hasse principle. Available at http://www.math.ubc.ca/$ \sim $dongquan/JTNB-algebraic-families.pdf.
  • [21] Benedict H. Gross, Hanoi lectures on the arithmetic of hyperelliptic curves, Acta Math. Vietnam. 37 (2012), no. 4, 579-588. MR 3058664
  • [22] Benedict H. Gross, On Bhargava's representation and Vinberg's invariant theory, Frontiers of mathematical sciences, Int. Press, Somerville, MA, 2011, pp. 317-321. MR 3050830
  • [23] Stephen Lichtenbaum, Duality theorems for curves over $ p$-adic fields, Invent. Math. 7 (1969), 120-136. MR 0242831
  • [24] John Milnor, Introduction to algebraic $ K$-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR 0349811
  • [25] Jin Nakagawa, Binary forms and orders of algebraic number fields, Invent. Math. 97 (1989), no. 2, 219-235. MR 1001839, https://doi.org/10.1007/BF01389040
  • [26] Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109-1149. MR 1740984, https://doi.org/10.2307/121064
  • [27] Bjorn Poonen and Michael Stoll, A local-global principle for densities, Topics in number theory (University Park, PA, 1997) Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 241-244. MR 1691323
  • [28] Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141-188. MR 1465369
  • [29] M. Reid, The complete intersection of two or more quadrics, Ph.D. Thesis, Trinity College, Cambridge (1972).
  • [30] Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris, 1959 (French). MR 0103191
  • [31] A. Shankar and X. Wang, Average size of the 2-Selmer group for monic even hyperelliptic curves, http://arxiv.org/abs/1307.3531.
  • [32] Samir Siksek, Chabauty for symmetric powers of curves, Algebra Number Theory 3 (2009), no. 2, 209-236. MR 2491943, https://doi.org/10.2140/ant.2009.3.209
  • [33] A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 114, 2007.
  • [34] Michael Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), no. 4, 349-391. MR 2368954, https://doi.org/10.2140/ant.2007.1.349
  • [35] Michael Stoll and Ronald van Luijk, Explicit Selmer groups for cyclic covers of $ \mathbb{P}^1$, Acta Arith. 159 (2013), no. 2, 133-148. MR 3062912, https://doi.org/10.4064/aa159-2-4
  • [36] Quôć Thańg Nguyêñ, Weak corestriction principle for non-abelian Galois cohomology, Homology, Homotopy Appl. 5 (2003), no. 1, 219-249. MR 2006400
  • [37] X. Wang, Maximal linear spaces contained in the base loci of pencils of quadrics, http://arxiv.org/abs/1302.2385.
  • [38] Xiaoheng Wang, Pencils of quadrics and Jacobians of hyperelliptic curves, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.), Harvard University. MR 3167287
  • [39] Melanie Matchett Wood, Rings and ideals parameterized by binary $ n$-ic forms, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 208-231. MR 2763952, https://doi.org/10.1112/jlms/jdq074
  • [40] Melanie Matchett Wood, Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math. 689 (2014), 169-199. MR 3187931, https://doi.org/10.1515/crelle-2012-0058

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11G30, 14G05

Retrieve articles in all journals with MSC (2000): 11G30, 14G05


Additional Information

Manjul Bhargava
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: bhargava@math.princeton.edu

Benedict H. Gross
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: gross@math.harvard.edu

Xiaoheng Wang
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: xw5@math.princeton.edu

Tim Dokchitser
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: tim.dokchitser@bristol.ac.uk

Vladimir Dokchitser
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: v.dokchitser@warwick.ac.uk

DOI: https://doi.org/10.1090/jams/863
Keywords: Rational points, hyperelliptic curves, Brauer-Manin obstruction, generalized Jacobian, points over extensions
Received by editor(s): November 14, 2013
Received by editor(s) in revised form: December 31, 2015, and April 20, 2016
Published electronically: July 27, 2016
Additional Notes: The first and third authors were supported by a Simons Investigator Grant and NSF grant DMS-1001828.
The second author was supported by NSF grant DMS-0901102.
The authors of the appendix were supported by Royal Society University Research Fellowships.
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society