Hanoi lectures on the arithmetic of hyperelliptic curves
نویسندگان
چکیده
Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average rank of the Mordell-Weil groups of their Jacobians over Q. As a consequence, we can use Chabauty’s method to obtain a uniform bound on the number of points on a majority of these curves, when the genus is at least 2.
منابع مشابه
Chabauty and the Mordell-Weil Sieve
These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.
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