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

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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.

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Acknowledgements

The second author is an International Research Fellow of the Japan Society for the Promotion of Science and is supported by a Grant-in-Aid for Scientific Research.

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Correspondence to Ehud Moshe Baruch.

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Baruch, E.M., Purkait, S. Hecke algebras, new vectors and new forms on \(\Gamma _0(m)\) . Math. Z. 287, 705–733 (2017). https://doi.org/10.1007/s00209-017-1842-y

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  • DOI: https://doi.org/10.1007/s00209-017-1842-y

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