Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.00711 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916657031544832 |
|---|---|
| author | Pajaziti, Antigona Sadek, Mohammad |
| author_facet | Pajaziti, Antigona Sadek, Mohammad |
| contents | Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00711 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Divisibility of orders of reductions of elliptic curves Pajaziti, Antigona Sadek, Mohammad Number Theory Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers. |
| title | Divisibility of orders of reductions of elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2301.00711 |