Big images of two-dimensional pseudorepresentations
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866916495311765504 |
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| author | Conti, Andrea Lang, Jaclyn Medvedovsky, Anna |
| author_facet | Conti, Andrea Lang, Jaclyn Medvedovsky, Anna |
| contents | Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $ρ$ of a profinite group on a local pro-$p$ domain $A$ contains a nontrivial congruence subgroup of ${\rm SL}_2(B)$ for a certain subring $B$ of $A$. We enlarge Bellaïche's ring and give this new $B$ a conceptual interpretation in terms of conjugate self-twists of $ρ$, symmetries that naturally constrain its image. As a corollary, this new $B$ is optimal among congruence subgroups contained in the image. We also interpret the new $B$ vis-a-vis the adjoint trace ring of $ρ$, which we show is a more natural ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for ${\rm GL}_2$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or Coleman families of elliptic and Hilbert modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_10519 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Big images of two-dimensional pseudorepresentations Conti, Andrea Lang, Jaclyn Medvedovsky, Anna Number Theory 11F11, 11F80, 11F85 Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $ρ$ of a profinite group on a local pro-$p$ domain $A$ contains a nontrivial congruence subgroup of ${\rm SL}_2(B)$ for a certain subring $B$ of $A$. We enlarge Bellaïche's ring and give this new $B$ a conceptual interpretation in terms of conjugate self-twists of $ρ$, symmetries that naturally constrain its image. As a corollary, this new $B$ is optimal among congruence subgroups contained in the image. We also interpret the new $B$ vis-a-vis the adjoint trace ring of $ρ$, which we show is a more natural ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for ${\rm GL}_2$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or Coleman families of elliptic and Hilbert modular forms. |
| title | Big images of two-dimensional pseudorepresentations |
| topic | Number Theory 11F11, 11F80, 11F85 |
| url | https://arxiv.org/abs/1904.10519 |