The average Mordell-Weil rank of elliptic surfaces over number fields

2022/04/26 by Kloosterman, Remke
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2204.12102

Abstract

Let K be a finitely generated field over ℚ. Let X→ B be a family of elliptic surfaces over K such that each elliptic fibration has the same configuration of singular fibers. Let r be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside |B| where the Mordell-Weil rank is at least r+1 is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over ℚ and prove a similar result for elliptic surfaces over arbitrary number fields.

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