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**A positive proportion of locally soluble hyperelliptic curves over \(\mathbb {Q}\) have no point over any odd degree extension.**
*(English)*
Zbl 1385.11043

Let \(C\) be a hyperelliptic curve over \(\mathbb{Q}\) of genus \(g\), expressed by an equation of the form \(z^2 = f(x,y)\) where \(f(x,y)\) is a homogeneuous polynomial of degree \(n=2g+2\) with integer coefficients \(a_0,a_1,\dots,a_n\), and \(f\) factors into distinct linear factors over \(\overline{\mathbb{Q}}\), and let the height of \(C\) be \(H(C):=\max\{ | a_k| : k = 0,\dots,n \}\). Given a positive integer \(X\) there are finitely many curves \(C\) with \(H(C)\leq X\), and the results of this paper use the enumeration determined by the finiteness of the number of curves with bounded height.

If \(C\) is not soluble, it does not have a rational point, and a harder question to answer is the rational points of \(C\) that are locally soluble. It is known [B. Poonen and M. Stoll, in: Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31–August 3, 1997. Dordrecht: Kluwer Academic Publishers. 241–244 (1999; Zbl 1024.11047)] that for fixed genus \(g\) more than 75% of the curves are locally soluble. In this paper, the authors prove that given genus \(g\) a positive proportion of locally soluble hyperelliptic curves \(C\) of genus \(g\) contain no rational points over any field extensions of \(\mathbb{Q}\) of odd degree \(\leq m\), and that given a field extension degree \(m\) a proportion of hyperelliptic curves \(C\) of genus \(g\) over \(\mathbb{Q}\) containing no points over all extensions of \(\mathbb{Q}\) of odd degree \(\leq m\) approaches \(1\) as \(g\to \infty\).

The authors’ approach is focused around the principal homogeneous space \(J^1=\mathrm{Pic}^1_{C/\mathbb{Q}}\) for \(J\). A point on \(C\) defined over an extension field of \(\mathbb{Q}\) of odd degree gives a rational point on \(J^1\), and they prove that there is a positive proportion of locally soluble hyperelliptic curves such that a certain Selmer set \(\mathrm{Sel}_2(J^1)\) is empty, which in turn implies that \(C \) has no rational points. The authors also introduce a similar statistical result on the Brauer-Manin obstructions to \(J^1\) having a rational point.

If \(C\) is not soluble, it does not have a rational point, and a harder question to answer is the rational points of \(C\) that are locally soluble. It is known [B. Poonen and M. Stoll, in: Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31–August 3, 1997. Dordrecht: Kluwer Academic Publishers. 241–244 (1999; Zbl 1024.11047)] that for fixed genus \(g\) more than 75% of the curves are locally soluble. In this paper, the authors prove that given genus \(g\) a positive proportion of locally soluble hyperelliptic curves \(C\) of genus \(g\) contain no rational points over any field extensions of \(\mathbb{Q}\) of odd degree \(\leq m\), and that given a field extension degree \(m\) a proportion of hyperelliptic curves \(C\) of genus \(g\) over \(\mathbb{Q}\) containing no points over all extensions of \(\mathbb{Q}\) of odd degree \(\leq m\) approaches \(1\) as \(g\to \infty\).

The authors’ approach is focused around the principal homogeneous space \(J^1=\mathrm{Pic}^1_{C/\mathbb{Q}}\) for \(J\). A point on \(C\) defined over an extension field of \(\mathbb{Q}\) of odd degree gives a rational point on \(J^1\), and they prove that there is a positive proportion of locally soluble hyperelliptic curves such that a certain Selmer set \(\mathrm{Sel}_2(J^1)\) is empty, which in turn implies that \(C \) has no rational points. The authors also introduce a similar statistical result on the Brauer-Manin obstructions to \(J^1\) having a rational point.

Reviewer: Sungkon Chang (Savannah)

### Citations:

Zbl 1024.11047
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\textit{M. Bhargava} et al., J. Am. Math. Soc. 30, No. 2, 451--493 (2017; Zbl 1385.11043)

### References:

[1] | B M.Bhargava, Most hyperelliptic curves over \(Q\) have no rational points, http://arxiv.org/abs/1308.0395v1. |

[2] | geosieve M.Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, http://arxiv.org/abs/1402.0031v1. |

[3] | BCF M.Bhargava, J.Cremona, and T.Fisher, The density of hyperelliptic curves over \(Q\) of genus \(g\) that have points everywhere locally, preprint. |

[4] | Bhargava, Manjul; Gross, Benedict H., Arithmetic invariant theory. Symmetry: representation theory and its applications, Progr. Math. 257, 33\textendash 54 pp. (2014), Birkh\"auser/Springer, New York · Zbl 1377.11045 |

[5] | Bhargava, Manjul; Gross, Benedict H., 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. 22, 23\textendash 91 pp. (2013), Tata Inst. Fund. Res., Mumbai · Zbl 1303.11072 |

[6] | AITII 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] | Bhargava, Manjul; Shankar, Arul, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2), 181, 1, 191\textendash 242 pp. (2015) · Zbl 1307.11071 |

[8] | Bhargava, Manjul; Shankar, Arul, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2), 181, 2, 587\textendash 621 pp. (2015) · Zbl 1317.11038 |

[9] | Birch, B. J.; Merriman, J. R., Finiteness theorems for binary forms with given discriminant, Proc. Lond. Math. Soc. (3), 24, 385\textendash 394 pp. (1972) · Zbl 0248.12002 |

[10] | BLR S.Bosch, W.L\`“utkebohmert, and M.Raynaud, N\'”eron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Berlin, New York: Springer-Verlag, 1990. |

[11] | Bruin, Nils; Stoll, Michael, Two-cover descent on hyperelliptic curves, Math. Comp., 78, 268, 2347\textendash 2370 pp. (2009) · Zbl 1208.11078 |

[12] | Cassels, J. W. S., The Mordell-Weil group of curves of genus \(2\). Arithmetic and geometry, Vol. I, Progr. Math. 35, 27\textendash 60 pp. (1983), Birkh\"auser, Boston, Mass. |

[13] | Colliot-Th{\'e}l{\`e}ne, Jean-Louis; Poonen, Bjorn, Algebraic families of nonzero elements of Shafarevich-Tate groups, J. Amer. Math. Soc., 13, 1, 83\textendash 99 pp. (2000) · Zbl 0951.11022 |

[14] | Colliot-Th{\'e}l{\`“e}ne, Jean-Louis; Sansuc, Jean-Jacques, La descente sur les vari\'”et\'es rationnelles. II, Duke Math. J., 54, 2, 375\textendash 492 pp. (1987) · Zbl 0659.14028 |

[15] | Dembo, Amir; Poonen, Bjorn; Shao, Qi-Man; Zeitouni, Ofer, Random polynomials having few or no real zeros, J. Amer. Math. Soc., 15, 4, 857\textendash 892 (electronic) pp. (2002) · Zbl 1002.60045 |

[16] | Desale, U. V.; Ramanan, S., Classification of vector bundles of rank \(2\) on hyperelliptic curves, Invent. Math., 38, 2, 161\textendash 185 pp. (1976/77) · Zbl 0323.14012 |

[17] | Dokchitser, Tim; Dokchitser, Vladimir, Self-duality of Selmer groups, Math. Proc. Cambridge Philos. Soc., 146, 2, 257\textendash 267 pp. (2009) · Zbl 1205.11065 |

[18] | Dokchitser, Tim; Dokchitser, Vladimir, Regulator constants and the parity conjecture, Invent. Math., 178, 1, 23\textendash 71 pp. (2009) · Zbl 1219.11083 |

[19] | Donagi, Ron, Group law on the intersection of two quadrics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7, 2, 217\textendash 239 pp. (1980) · Zbl 0457.14023 |

[20] | NDQ 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. · Zbl 1341.14008 |

[21] | Gross, Benedict H., Hanoi lectures on the arithmetic of hyperelliptic curves, Acta Math. Vietnam., 37, 4, 579\textendash 588 pp. (2012) · Zbl 1294.11107 |

[22] | Gross, Benedict H., On Bhargava’s representation and Vinberg’s invariant theory. Frontiers of mathematical sciences, 317\textendash 321 pp. (2011), Int. Press, Somerville, MA |

[23] | Lichtenbaum, Stephen, Duality theorems for curves over \(p\)-adic fields, Invent. Math., 7, 120\textendash 136 pp. (1969) · Zbl 0186.26402 |

[24] | Milnor, John, Introduction to algebraic \(K\)-theory, xiii+184 pp. (1971), Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo · Zbl 0237.18005 |

[25] | Nakagawa, Jin, Binary forms and orders of algebraic number fields, Invent. Math., 97, 2, 219\textendash 235 pp. (1989) · Zbl 0647.10018 |

[26] | Poonen, Bjorn; Stoll, Michael, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2), 150, 3, 1109\textendash 1149 pp. (1999) · Zbl 1024.11040 |

[27] | Poonen, Bjorn; Stoll, Michael, A local-global principle for densities. Topics in number theory, University Park, PA, 1997, Math. Appl. 467, 241\textendash 244 pp. (1999), Kluwer Acad. Publ., Dordrecht · Zbl 1024.11047 |

[28] | Poonen, Bjorn; Schaefer, Edward F., Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math., 488, 141\textendash 188 pp. (1997) · Zbl 0888.11023 |

[29] | R M.Reid, The complete intersection of two or more quadrics, Ph.D. Thesis, Trinity College, Cambridge (1972). |

[30] | Serre, Jean-Pierre, Groupes alg\'ebriques et corps de classes, 202 pp. (1959), Publications de l’institut de math\'ematique de l’universit\'e de Nancago, VII. Hermann, Paris · Zbl 0097.35604 |

[31] | SW A.Shankar and X.Wang, Average size of the 2-Selmer group for monic even hyperelliptic curves, http://arxiv.org/abs/1307.3531. |

[32] | Siksek, Samir, Chabauty for symmetric powers of curves, Algebra Number Theory, 3, 2, 209\textendash 236 pp. (2009) · Zbl 1254.11065 |

[33] | Sk A.Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 114, 2007. |

[34] | Stoll, Michael, Finite descent obstructions and rational points on curves, Algebra Number Theory, 1, 4, 349\textendash 391 pp. (2007) · Zbl 1167.11024 |

[35] | Stoll, Michael; van Luijk, Ronald, Explicit Selmer groups for cyclic covers of \(\mathbb{P}^1\), Acta Arith., 159, 2, 133\textendash 148 pp. (2013) · Zbl 1316.11056 |

[36] | Nguy{\^e}{\~n}, Qu{\^o}{\'c} Th{\v{a}}{\'n}g, Weak corestriction principle for non-abelian Galois cohomology, Homology, Homotopy Appl., 5, 1, 219\textendash 249 pp. (2003) · Zbl 1065.11021 |

[37] | W X.Wang, Maximal linear spaces contained in the base loci of pencils of quadrics, http://arxiv.org/abs/1302.2385. · Zbl 1419.14042 |

[38] | Wang, Xiaoheng, Pencils of quadrics and Jacobians of hyperelliptic curves, 148 pp. (2013), ProQuest LLC, Ann Arbor, MI |

[39] | Wood, Melanie Matchett, Rings and ideals parameterized by binary \(n\)-ic forms, J. Lond. Math. Soc. (2), 83, 1, 208\textendash 231 pp. (2011) · Zbl 1228.11053 |

[40] | Wood, Melanie Matchett, Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math., 689, 169\textendash 199 pp. (2014) · Zbl 1317.11039 |

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