On the average exponent of elliptic curves modulo $p$

Details On the average exponent of elliptic curves modulo $p$ On the average exponent of elliptic curves modulo $p$

2012-03-20

arxiv.org/abs/1203.4382v2

Mathematics

Number Theory

Section 7 added. Proof of Lemma 3.5 corrected. Other minor corrections

Scientific paper

Given an elliptic curve $E$ defined over $\mathbb{Q}$ and a prime $p$ of good reduction, let $\tilde{E}(\mathbb{F}_p)$ denote the group of $\mathbb{F}_p$-points of the reduction of $E$ modulo $p$, and let $e_p$ denote the exponent of said group. Assuming a certain form of the Generalized Riemann Hypothesis (GRH), we study the average of $e_p$ as $p \le X$ ranges over primes of good reduction, and find that the average exponent essentially equals $p\cdot c_{E}$, where the constant $c_{E} > 0$ depends on $E$. For $E$ without complex multiplication (CM), $c_{E}$ can be written as a rational number (depending on $E$) times a universal constant. Without assuming GRH, we can determine the average exponent when $E$ has CM, as well as give an upper bound on the average in the non-CM case.

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