TY - JOUR

T1 - Rational points on hyperelliptic curves having a marked non-Weierstrass point

AU - Shankar, Arul

AU - Wang, Xiaoheng

N1 - Funding Information:
We are very grateful to Manjul Bhargava and Benedict Gross for suggesting this problem to us and for many helpful conversations. We are also very grateful to Bjorn Poonen for explaining Chabauty’s method to us and for helpful comments on earlier versions of the argument. We are extremely grateful to Cheng-Chiang Tsai, Jacob Tsimerman, and Ila Varma for several helpful conversations. We are also very grateful for detailed and helpful comments from the annonymous referee. The first author is grateful for support from NSF grant DMS-1128155. The second author is grateful for support from a Simons Investigator Grant and NSF grant DMS-1001828.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this paper, we consider the family of hyperelliptic curves over Q having a fixed genus n and a marked rational non-Weierstrass point. We show that when n > 9, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as n tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of 5=2 on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137-1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and L-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23-91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

AB - In this paper, we consider the family of hyperelliptic curves over Q having a fixed genus n and a marked rational non-Weierstrass point. We show that when n > 9, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as n tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of 5=2 on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137-1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and L-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23-91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

KW - Selmer groups

KW - hyperelliptic curves

KW - ranks of abelian varieties

KW - rational points on curves

UR - http://www.scopus.com/inward/record.url?scp=85032924729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032924729&partnerID=8YFLogxK

U2 - 10.1112/S0010437X17007515

DO - 10.1112/S0010437X17007515

M3 - Article

AN - SCOPUS:85032924729

VL - 154

SP - 188

EP - 222

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 1

ER -