Modular curves and modular representations of SL2

Bryden Cais

Fix a prime p and let X(p) be the modular curve over the integers classifying elliptic curves with full-level p structure. The group G := SL2(Fp) acts on X(p) and hence on its (sheaf ) cohomology. In this talk, we we will investigate the structure of the Z[G]-module M given by the global sections of the canonical sheaf. In particular, we will describe the reduction modulo p of M as a mod p (modular) representations of G. This description relies heavily on the geometry of X(p) in characteristic p and uses Rosenlicht's description of the dualizing sheaf in terms of regular differential forms.