A local-global principle for polyquadratic twists of abelian surfaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911764041433088 |
|---|---|
| author | Fité, Francesc Perucca, Antonella |
| author_facet | Fité, Francesc Perucca, Antonella |
| contents | We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fité, relying itself on the works by Rajan and Ramakrishnan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06317 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A local-global principle for polyquadratic twists of abelian surfaces Fité, Francesc Perucca, Antonella Number Theory 11G10, 11G35 We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fité, relying itself on the works by Rajan and Ramakrishnan. |
| title | A local-global principle for polyquadratic twists of abelian surfaces |
| topic | Number Theory 11G10, 11G35 |
| url | https://arxiv.org/abs/2210.06317 |