We consider the family of hyperelliptic curves over Q of fixed genus along with a marked rational Weierstrass point and a marked rational nonWeierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2, and that most such curves have only three rational points. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also consider another related family of curves and study the interplay between these two families. There is a family φ of isogenies between these two families, and we prove that the average size of the φ-Selmer groups is exactly 2.
ASJC Scopus subject areas
- Applied Mathematics