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Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroupΓ  ⁢   (  p  ){\Gamma(p)}   ,  p  a prime, is acted on bySL  2   ⁢   (p   ){\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 bySL  2   ⁢   (p   ){\mathrm{SL}_{2}(\mathbb{F}_{p})}   . In this note, we compute the relation between these two spaces in the weight 2 case.

A note on cusp forms and representations of SL 2 (𝔽 𝑝 )