Modularity via Serre weights Sample Clauses

Modularity via Serre weights. Given a Xxxxxxx modular cusp- form f , which is an eigenform for the Hecke operators, Theorem 3.1.3 as- sociates to f a Galois representation ρf . By taking a lattice fixed by this representation, reducing modulo p and taking the semisimplification, we obtain a well-defined mod p representation ρf : Gal(F /F ) → GL2(Fp) associated to f . One then expects the following “folklore” generalisation of Serre’s conjecture to hold.
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