On the average number of 2-Selmer elements of elliptic curves over \mathbbF_q(X) with two marked points

2016/07/04 by Jack A. Thorne, Thorne, Jack A. · 1 citation
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Historical Studies and Socio-cultural Analysis

paper · pdf · doi:10.48550/arxiv.1607.00997

Abstract

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of G-torsors over an algebraic curve, where G is isogenous to SL₂⁴, and show that the images of 2-Selmer elements under this map become equidistributed in the limit.

Cited by

• Families of curves in Vinberg representations

Related

• The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
• 2-Selmer Groups over Multiquadratic Extensions
• On average sizes of Selmer groups and ranks in families of elliptic curves having marked points
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• Average size of 2-Selmer groups of Jacobians of hyperelliptic curves over function fields