Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2511.11948 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918223539077120 |
|---|---|
| author | Couvillon, Zachary Ray, Anwesh |
| author_facet | Couvillon, Zachary Ray, Anwesh |
| contents | We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from certain genus $0$ modular curves. We apply techniques from the geometry of numbers and sieve methods to prove that the number of elliptic curves with unexplained entanglements $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[3]) \neq \mathbb{Q}$ and $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[5]) \neq \mathbb{Q}$ and naive height $\leq X$, grows as $\gg X^{1/9}$ and $\gg X^{1/12}$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting elliptic curves with prescribed entanglements Couvillon, Zachary Ray, Anwesh Number Theory 11G05, 11R45, 11P21 We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from certain genus $0$ modular curves. We apply techniques from the geometry of numbers and sieve methods to prove that the number of elliptic curves with unexplained entanglements $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[3]) \neq \mathbb{Q}$ and $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[5]) \neq \mathbb{Q}$ and naive height $\leq X$, grows as $\gg X^{1/9}$ and $\gg X^{1/12}$, respectively. |
| title | Counting elliptic curves with prescribed entanglements |
| topic | Number Theory 11G05, 11R45, 11P21 |
| url | https://arxiv.org/abs/2511.11948 |