Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2205.05062 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914365820633088 |
|---|---|
| author | Whitmore, Dmitri |
| author_facet | Whitmore, Dmitri |
| contents | We construct a local deformation problem for residual Galois representations $\barρ$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of $\hat{G}$-adequate subgroup, our corresponding 'big image' condition. When $\hat{G}$ is a simply connected simple group of type $\mathrm{C}$ or of exceptional type and $\hat{G} \to \mathrm{GL}_n$ is a faithful irreducible representation of minimal dimension, we show that a subgroup is $\hat{G}$-adequate if it is $\mathrm{GL}_n$-irreducible and the residue characteristic is sufficiently large.
We apply our ideas to the case $\hat{G} = \mathrm{GSp}_4$ and prove a modularity lifting theorem for abelian surfaces over a totally real field $F$ which holds under weaker hypotheses than in the work of Boxer-Calegari-Gee-Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of $F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05062 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Taylor-Wiles method for reductive groups Whitmore, Dmitri Number Theory We construct a local deformation problem for residual Galois representations $\barρ$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of $\hat{G}$-adequate subgroup, our corresponding 'big image' condition. When $\hat{G}$ is a simply connected simple group of type $\mathrm{C}$ or of exceptional type and $\hat{G} \to \mathrm{GL}_n$ is a faithful irreducible representation of minimal dimension, we show that a subgroup is $\hat{G}$-adequate if it is $\mathrm{GL}_n$-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case $\hat{G} = \mathrm{GSp}_4$ and prove a modularity lifting theorem for abelian surfaces over a totally real field $F$ which holds under weaker hypotheses than in the work of Boxer-Calegari-Gee-Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of $F$. |
| title | The Taylor-Wiles method for reductive groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2205.05062 |