The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2

2023/10/12 by Achenjang, Niven
#11G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2310.08493

Abstract

Let K be the function field of a smooth curve B over a finite field k of arbitrary characteristic. We prove that the average size of the 2-Selmer groups of elliptic curves E/K is at most 1+2ζ_B(2)ζ_B(10), where ζ_B is the zeta function of the curve B. In particular, in the limit as q=#k→∞ (with the genus g(B) fixed), we see that the average size of 2-Selmer is bounded above by 3, even in "bad" characteristics. This completes the proof that the average rank of elliptic curves, over any fixed global field, is finite. Handling the case of characteristic 2 requires us to develop a new theory of integral models of 2-Selmer elements, dubbed "hyper-Weierstrass curves."

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