The average analytic rank of elliptic curves with prescribed torsion

Abstract

We show that the average analytic rank of elliptic curves with prescribed torsion G$G$ is bounded for every torsion group G$G$ under Generalized Riemann Hypothesis (GRH) for elliptic curve L$L$‐functions and, in addition, some moment conditions for G=double-struckZ/ndouble-struckZ$G = \mathbb {Z}/n\mathbb {Z}$, n0.16em=0.16em7,8,9,10,12$n \,{=}\, 7, 8,9,10,12$, and double-struckZ/2double-struckZ×double-struckZ/2mdouble-struckZ$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2m\mathbb {Z}$, m=3,4$m = 3, 4$.