Conductor exponents for families of hyperelliptic curves
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910673335746560 |
|---|---|
| author | Azon, Martin Curcó-Iranzo, Mar Khawaja, Maleeha Maistret, Céline Mocanu, Diana |
| author_facet | Azon, Martin Curcó-Iranzo, Mar Khawaja, Maleeha Maistret, Céline Mocanu, Diana |
| contents | We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conductor exponents for families of hyperelliptic curves Azon, Martin Curcó-Iranzo, Mar Khawaja, Maleeha Maistret, Céline Mocanu, Diana Number Theory 11D41, 11G30, 11G20, 14Q05, 11F80, 14H20 We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families. |
| title | Conductor exponents for families of hyperelliptic curves |
| topic | Number Theory 11D41, 11G30, 11G20, 14Q05, 11F80, 14H20 |
| url | https://arxiv.org/abs/2410.21134 |