Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup Γ⁢(p){\Gamma(p)}, p a prime, is acted on by SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})}. Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})}. In this note, we compute the relation between these two spaces in the weight 2 case.

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关于SL2（𝔽𝑝）的尖角形式和表示的注释

尖峰形式是在上半平面上定义的某些全纯函数，并且主要同余子组Γ⁢（p）{\ Gamma（p）}（prime素）的尖峰形式的空间受SL2⁢（𝔽p）的作用{\ mathrm {SL} _ {2}（\ mathbb {F} _ {p}）}。同时，上半平面的有限场化身为Deligne-Lusztig（或Drinfeld）曲线，其同调空间也受SL2⁢（𝔽p）{\ mathrm {SL} _ {2}（\ mathbb {F} _ {p}）}。在本说明中，我们在权重为2的情况下计算这两个空间之间的关系。