K-varieties and Galois representations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912143733948416 |
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| author | Florit, Enric Pacetti, Ariel |
| author_facet | Florit, Enric Pacetti, Ariel |
| contents | In a remarkable article Ribet showed how to attach rational $2$-dimensional representations to elliptic ${\mathbb Q}$-curves. An abelian variety $A$ is a (weak) $K$-variety if it is isogenous to all of its $\text{Gal}_K$-conjugates. In this article we study the problem of attaching an absolutely irreducible $\ell$-adic representation of $\text{Gal}_K$ to an abelian $K$-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over ${\mathbb Q}$ with potential quaternionic multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-varieties and Galois representations Florit, Enric Pacetti, Ariel Number Theory In a remarkable article Ribet showed how to attach rational $2$-dimensional representations to elliptic ${\mathbb Q}$-curves. An abelian variety $A$ is a (weak) $K$-variety if it is isogenous to all of its $\text{Gal}_K$-conjugates. In this article we study the problem of attaching an absolutely irreducible $\ell$-adic representation of $\text{Gal}_K$ to an abelian $K$-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over ${\mathbb Q}$ with potential quaternionic multiplication. |
| title | K-varieties and Galois representations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.03184 |