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Hecke Algebras, New Vectors and New Spaces

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The Computational and Theoretical Aspects of Elliptic Curves

Part of the book series: Mathematical Lectures from Peking University ((MLPKU))

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Abstract

We report a joint work [2] with E. M. Baruch in which we characterize the space of new forms for \(\Gamma _0(N)\) as a common eigenspace of certain Hecke operators which depend on primes dividing the level N.

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References

  1. A.O.L. Atkin, J. Lehner, Hecke operators on \(\Gamma _0(m)\). Math. Ann. 185, 134–160 (1970)

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

  3. W. Casselman, On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973)

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  4. W. Casselman, The restriction of a representation of GL\(_2(k)\) to GL\(_2(\mathfrak{o})\). Math. Ann. 206, 311–318 (1973)

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  5. S. Gelbart, Automorphic forms on adele groups. Annals of Mathematics Studies, vol. 83 (Princeton University Press, Princeton, 1975)

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  6. R. Howe, Affine-like Hecke algebras and \(p\) -adic representation theory, in Iwahori-Hecke Algebras and their Representation Theory. Lecture Notes in Mathematics, vol. 1804 (2002), pp. 27–69

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Acknowledgements

It was a pleasure to attend the first China-Indo conference on Elliptic curves in the BICMR where I presented these results. I am grateful for the invitation.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Tokyo, Japan

    Soma Purkait

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  1. Soma Purkait
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Correspondence to Soma Purkait .

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Editors and Affiliations

  1. Capital Normal University, Beijing, China

    Zhibin Liang

  2. Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Sahibzada Ajit Singh Nagar, Punjab, India

    Chandrakant Aribam

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Purkait, S. (2019). Hecke Algebras, New Vectors and New Spaces. In: Liang, Z., Aribam, C. (eds) The Computational and Theoretical Aspects of Elliptic Curves. Mathematical Lectures from Peking University. Springer, Singapore. https://doi.org/10.1007/978-981-13-6664-2_6

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