论文标题
泰勒 - 韦尔斯的还原组
The Taylor-Wiles method for reductive groups
论文作者
论文摘要
我们为剩余的Galois表示形式构建了一个局部变形问题$ \barρ$，该$ \barρ$在任意还原组$ \ hat {g} $中，我们用来开发Taylor-Wiles方法的变体。我们的概括允许泰勒 - 韦尔斯（Taylor-Wiles）的位置，弗罗贝尼乌斯（Frobenius）的图像是半神经，这是文献中先前施加的常规半神经约束的削弱。我们介绍了$ \ hat {g} $的概念 - 足够的子组，我们对应的“大图像”条件。当$ \ hat {g} $是一个简单连接的简单类型的$ \ mathrm {c} $或异常类型和$ \ hat {g} \ to \ mathrm {gl} _n $是一个忠实的不可约束的尺寸，我们表明子组是$ \ hat IS if at iS if-fat if-ad iS-if-ad iS- $ \ mathrm {gl} _n $ -rirreducible，残基特性足够大。 我们将我们的想法应用于案例$ \ hat {g} = \ mathrm {gsp} _4 $，并在完全真实的领域$ f $上证明了一个模块化的Abelian表面定理，该$ F $比Boxer-Calegari-Gee-Gee-Pilloni的工作较弱。我们推断出椭圆曲线的一些模块化结果，而不是$ f $的二次扩展。
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 $F$.
