论文标题
过度会议的模块化形式是完成的共同体的Hodge-Tate重量零部分中的最高权重向量
Overconvergent modular forms are highest weight vectors in the Hodge-Tate weight zero part of completed cohomology
论文作者
论文摘要
我们构建了$（\ Mathfrak {gl} _2，b（\ Mathbb {q} _p））$和Hecke-Equivariant杯子产品配对，与$ 0 $的$ 0 $在$ \ mathbb {p}^1 $中，$ 0 $的$ 0 $之间的$ 0 $在$ 0. $ \ mathbb {c} _p $ - 模块化曲线的火药学。本地的同一个同种学组是一个最大的Verma模块，对于任何过度会变化的模块化模块化形式的无穷小重量不等于$ 1 $，杯子的重量向量最高。对于经典的重量$ k \ geq 2 $，Verma具有代数的$ h^1（\ Mathbb {p}^1，\ Mathcal {o}（ - k））$，并且在经典形式上通过此商来提供配对因素，从而对局部代理人的几何描述进行了几何描述。另一半由与$ h^1 $的角色配对，$ h^0 $在模块化曲线和$ \ mathbb {p}^1 $之间反转。在较小的假设下，我们推断出与过度会议模块化形式的Galois表示的hodge-tate-sen重量的猜想。我们的主要结果本质上是由Arxiv：2008.07099独立于Lue Pan独立获得的部分的严格子集，但是这里的视角不同，证明是简短的，使用了简单的工具：Mayer-Vietoris覆盖物，Cup产品，杯子产品和组的边界图。
We construct a $(\mathfrak{gl}_2, B(\mathbb{Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb{P}^1$, landing in the compactly supported completed $\mathbb{C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb{P}^1, \mathcal{O}(-k))$, and on classical forms the pairing factors through this quotient, giving a geometric description of "half" of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb{P}^1$. Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan in arXiv:2008.07099, but the perspective here is different and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.
