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

HTML articles powered by AMS MathViewer

- by Manjul Bhargava, Benedict H. Gross and Xiaoheng Wang; with an appendix by Tim Dokchitser; with an appendix by Vladimir Dokchitser PDF
- J. Amer. Math. Soc.
**30**(2017), 451-493 Request permission

## 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

- M. Bhargava, Most hyperelliptic curves over ${\mathbb Q}$ have no rational points, http://arxiv.org/abs/1308.0395v1.
- M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, http://arxiv.org/abs/1402.0031v1.
- M. Bhargava, J. Cremona, and T. Fisher, The density of hyperelliptic curves over ${\mathbb Q}$ of genus $g$ that have points everywhere locally, preprint.
- 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**, DOI 10.1007/978-1-4939-1590-3_{3} - 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** - 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. - 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**, DOI 10.4007/annals.2015.181.1.3 - 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**, DOI 10.4007/annals.2015.181.2.4 - B. J. Birch and J. R. Merriman,
*Finiteness theorems for binary forms with given discriminant*, Proc. London Math. Soc. (3)**24**(1972), 385–394. MR**306119**, DOI 10.1112/plms/s3-24.3.385 - S. Bosch, W. Lütkebohmert, and M. Raynaud,
*Néron models*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3),**21**, Berlin, New York: Springer-Verlag, 1990. - Nils Bruin and Michael Stoll,
*Two-cover descent on hyperelliptic curves*, Math. Comp.**78**(2009), no. 268, 2347–2370. MR**2521292**, DOI 10.1090/S0025-5718-09-02255-8 - 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** - 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**, DOI 10.1090/S0894-0347-99-00315-X - 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**, DOI 10.1215/S0012-7094-87-05420-2 - 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. MR**1915821**, DOI 10.1090/S0894-0347-02-00386-7 - 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**429897**, DOI 10.1007/BF01408570 - Tim Dokchitser and Vladimir Dokchitser,
*Self-duality of Selmer groups*, Math. Proc. Cambridge Philos. Soc.**146**(2009), no. 2, 257–267. MR**2475965**, DOI 10.1017/S0305004108001989 - Tim Dokchitser and Vladimir Dokchitser,
*Regulator constants and the parity conjecture*, Invent. Math.**178**(2009), no. 1, 23–71. MR**2534092**, DOI 10.1007/s00222-009-0193-7 - Ron Donagi,
*Group law on the intersection of two quadrics*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**7**(1980), no. 2, 217–239. MR**581142** - 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.
- Benedict H. Gross,
*Hanoi lectures on the arithmetic of hyperelliptic curves*, Acta Math. Vietnam.**37**(2012), no. 4, 579–588. MR**3058664** - 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** - Stephen Lichtenbaum,
*Duality theorems for curves over $p$-adic fields*, Invent. Math.**7**(1969), 120–136. MR**242831**, DOI 10.1007/BF01389795 - John Milnor,
*Introduction to algebraic $K$-theory*, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR**0349811** - Jin Nakagawa,
*Binary forms and orders of algebraic number fields*, Invent. Math.**97**(1989), no. 2, 219–235. MR**1001839**, DOI 10.1007/BF01389040 - 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**, DOI 10.2307/121064 - 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** - 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** - M. Reid, The complete intersection of two or more quadrics, Ph.D. Thesis, Trinity College, Cambridge (1972).
- 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** - A. Shankar and X. Wang, Average size of the 2-Selmer group for monic even hyperelliptic curves, http://arxiv.org/abs/1307.3531.
- Samir Siksek,
*Chabauty for symmetric powers of curves*, Algebra Number Theory**3**(2009), no. 2, 209–236. MR**2491943**, DOI 10.2140/ant.2009.3.209 - A. Skorobogatov,
*Torsors and rational points*, Cambridge Tracts in Mathematics**114**, 2007. - Michael Stoll,
*Finite descent obstructions and rational points on curves*, Algebra Number Theory**1**(2007), no. 4, 349–391. MR**2368954**, DOI 10.2140/ant.2007.1.349 - Michael Stoll and Ronald van Luijk,
*Explicit Selmer groups for cyclic covers of $\Bbb P^1$*, Acta Arith.**159**(2013), no. 2, 133–148. MR**3062912**, DOI 10.4064/aa159-2-4 - Nguyêñ Quôć Thǎńg,
*Weak corestriction principle for non-abelian Galois cohomology*, Homology Homotopy Appl.**5**(2003), no. 1, 219–249. MR**2006400**, DOI 10.4310/HHA.2003.v5.n1.a10 - X. Wang, Maximal linear spaces contained in the base loci of pencils of quadrics, http://arxiv.org/abs/1302.2385.
- Xiaoheng Wang,
*Pencils of quadrics and Jacobians of hyperelliptic curves*, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Harvard University. MR**3167287** - 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**, DOI 10.1112/jlms/jdq074 - Melanie Matchett Wood,
*Parametrization of ideal classes in rings associated to binary forms*, J. Reine Angew. Math.**689**(2014), 169–199. MR**3187931**, DOI 10.1515/crelle-2012-0058

## Additional Information

**Manjul Bhargava**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 623882
- Email: bhargava@math.princeton.edu
**Benedict H. Gross**- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 77400
- Email: gross@math.harvard.edu
**Xiaoheng Wang**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 1074838
- Email: xw5@math.princeton.edu
**Tim Dokchitser**- Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- MR Author ID: 733080
- Email: tim.dokchitser@bristol.ac.uk
**Vladimir Dokchitser**- Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 768165
- Email: v.dokchitser@warwick.ac.uk
- 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. - © Copyright 2016 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 451-493 - MSC (2000): Primary 11G30; Secondary 14G05
- DOI: https://doi.org/10.1090/jams/863
- MathSciNet review: 3600041