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

TL;DR

This paper determines the asymptotic behavior of the average of the first invariant factor in the reduction of CM elliptic curves mod p, showing it is proportional to x for large x.

Contribution

It establishes the precise order of magnitude for the sum of the first invariant factors, confirming it is proportional to x for large x, refining previous bounds.

Findings

Sum of d_p over primes p up to x is proportional to x.

Confirms the order of magnitude of the average invariant factor for CM elliptic curves.

Refines previous bounds by establishing the asymptotic proportionality to x.

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 $x$.