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.
Publisher Copyright:
© 2017 Foundation Compositio Mathematica.
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
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U2 - 10.1112/S0010437X17007515
DO - 10.1112/S0010437X17007515
M3 - Article
AN - SCOPUS:85032924729
SN - 0010-437X
VL - 154
SP - 188
EP - 222
JO - Compositio Mathematica
JF - Compositio Mathematica
IS - 1
ER -