Hecke algebras, new vectors and new forms on $$\Gamma _0(m)$$ Γ 0 ( m )

Abstract

We characterize the space of new forms for \(\Gamma _0(m)\) as a common eigenspace of certain Hecke operators which depend on primes p dividing the level m. To do that we find generators and relations for a p-adic Hecke algebra of functions on \(K={{\mathrm{GL}}}_2({\mathbb {Z}}_p)\). We explicitly find the \(n+1\) irreducible representations of K which contain a vector of level n including the unique representation that contains the “new vector” of level n. After translating the p-adic Hecke operators that we obtain into classical Hecke operators we obtain the results about the new space mentioned above.