论文标题
较高尺寸的无限蕨
The infinite fern in higher dimensions
论文作者
论文摘要
如果$ \ \barρ$是一种自动模型$ P $ GALOIS表示，那么自然而然地想知道自动点点是否在$ \barρ$的变形空间中是Zariski致密。在$ p $的统一集团拆分（和未经夸大）的情况下，我们证明了这一方向的新结果。 Namely, if $\barρ$ is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of $\barρ$ is Zariski dense in a non-empty union of irreducible components.这特别是Gouvêa-Mazur的概括，以$ GL_2/\ MATHBB Q $，CHENEVIER $ U（3）$（3）$和最近的Hellmann-Margerin-Schraen。新颖性是，我们使用Breuil-Hellmann-Schraen的局部模型来控制局部变形环中的切线空间，以及最初是由于Bellaïche-Chenevier和Taïbi而引起的对特征变量的几何论点，以减少具有巨大图像的点。在这些地方，我们可以使用牛顿 - 索恩（Newton-Thorne）的最新结果来控制一个塞尔默（Selmer）群体的消失。特别是，我们不需要在$ \barρ$上假设任何“泰勒 - 韦尔斯”假设，这尤其是不可约束的。如果我们在$ \barρ$上添加泰勒 - 韦尔斯假设，并以$ p $添加额外的假设，则由于艾伦到处都是扎里斯基密度的结果。
If $\barρ$ is an automorphic modulo $p$ Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of $\barρ$. We prove new results in this direction in the case of a unitary group split (and unramified) at $p$. Namely, if $\barρ$ is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of $\barρ$ is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouvêa-Mazur for $GL_2/\mathbb Q$, Chenevier for $U(3)$ and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bellaïche-Chenevier and Taïbi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on $\barρ$, which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on $\barρ$ and an extra hypothesis at $p$, we have by a result of Allen the Zariski density everywhere.
