Local data of elliptic curves under quadratic twist
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909735915094016 |
|---|---|
| author | Barrios, Alexander J. Roy, Manami Sahajpal, Nandita Tallana, Darwin Tobin, Bella Wiersema, Hanneke |
| author_facet | Barrios, Alexander J. Roy, Manami Sahajpal, Nandita Tallana, Darwin Tobin, Bella Wiersema, Hanneke |
| contents | Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$. We note that when the residue field has characteristic $2$, we only consider the special case $K=\mathbb{Q}_{2}$. In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03209 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local data of elliptic curves under quadratic twist Barrios, Alexander J. Roy, Manami Sahajpal, Nandita Tallana, Darwin Tobin, Bella Wiersema, Hanneke Number Theory 11G07, 14H10, 14H52, 11G05 Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$. We note that when the residue field has characteristic $2$, we only consider the special case $K=\mathbb{Q}_{2}$. In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$. |
| title | Local data of elliptic curves under quadratic twist |
| topic | Number Theory 11G07, 14H10, 14H52, 11G05 |
| url | https://arxiv.org/abs/2501.03209 |