The average of the first invariant factor for reductions of CM elliptic curves mod $p$

Abstract

Let $E/\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E({\mathbb{F}}_p)\cong \mathbb{Z}/d_p\mathbb{Z}\oplus \mathbb{Z}/e_p\mathbb{Z}$, where $d_p\mid e_p$. Kowalski proposed investigating the average value of $d_p$ as $p$ runs over the rational primes. For CM curves, he showed that $x\log \log x/\log x\ll {\sum }_{p\le x}{d}_p\ll x\sqrt{\log x}$. It was shown recently by Felix and Murty that in fact ${\sum }_{p\le x}{d}_p$ exceeds any constant multiple of $x\log \log x/\log x$, once $x$ is sufficiently large. In the opposite direction, Kim has shown that the expression $x\sqrt{\log x}$ in the upper bound can be replaced by $x\log \log x$. In this paper, we obtain the correct order of magnitude for the sum: ${\sum }_{p\le x}{d}_p\asymp x$ for all large $x$.