## Modular curves and modular representations of **SL**_{2}

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 := **SL**_{2}(**F**_{p}) 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.

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