The average analytic rank of elliptic curves with prescribed torsion

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=Z/nZ$G = \mathbb {Z}/n\mathbb {Z}$, n=7,8,9,10,12$n \,{=}\, 7, 8,9,10,12$, and Z/2Z×Z/2mZ$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2m\mathbb {Z}$, m=3,4$m = 3, 4$.