The average number of subgroups of elliptic curves over finite fields

2018/11/26 by Perret-Gentil, Corentin
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.10149

Abstract

By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most 2. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić--Zhai and Blomer.

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