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- W4313484503 abstract "Let $E$ be an elliptic curve defined over $mathbb{Q}$ and $widetilde{E}_p(mathbb{F}_p)$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. Given an integer $mge 2$ and any $a$ modulo $m$, we consider how often the congruence $|widetilde{E}_p(mathbb{F}_p)|equiv amod m$ holds. We show that the greatest common divisor of the integers $|widetilde{E}_p(mathbb{F}_p)|$ over all rational primes $p$ cannot exceed $4$. We then exhibit elliptic curves over $mathbb{Q}(t)$ with trivial torsion for which the orders of reductions of every smooth fiber modulo primes of positive density at least $1/2$ are divisible by a fixed small integer. We also show that if the torsion of $E$ grows over a quadratic field $K$, then one may explicitly compute $|widetilde{E}_p(mathbb{F}_p)|$ modulo the torsion subgroup $|E(K)_{tors}|$. More precisely, we show that there exists an integer $Nge 2$ such that $|widetilde{E}_p(mathbb{F}_p)|$ is determined modulo $|E(K)_{tors}|$ according to the arithmetic progression modulo $N$ in which $p$ lies. It follows that given any $a$ modulo $|E(K)_{tors}|$, we can estimate the density of primes $p$ such that the congruence $|widetilde{E}_p(mathbb{F}_p)|equiv amod |E(K)_{tors}|$ occurs." @default.
- W4313484503 created "2023-01-06" @default.
- W4313484503 creator A5052438341 @default.
- W4313484503 creator A5055010854 @default.
- W4313484503 date "2023-01-02" @default.
- W4313484503 modified "2023-09-30" @default.
- W4313484503 title "On congruence classes of orders of reductions of elliptic curves" @default.
- W4313484503 doi "https://doi.org/10.48550/arxiv.2301.00711" @default.
- W4313484503 hasPublicationYear "2023" @default.
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