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The cohomology of modular curves as a module over the Galois group
Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
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Is there an analogue of theta cycles for more general mod p automorphic forms?
The theory of $\theta$-cycles (due to Tate, I think) and filtrations is to me a very beautiful and powerful tool in proving many statements about mod $p$ modular forms in more explicit and elementary ...
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