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On the classification of lattices over \(\mathbb{Q}(\sqrt{-3})\), which are even unimodular ℤ-lattices

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References

  1. Bachoc, C., Nebe, G.: Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order. Exp. Math. 6, 151–162 (1997)

    MATH  MathSciNet  Google Scholar 

  2. Borcherds, R.E., Freitag, E., Weissauer, R.: A Siegel cusp form of degree 12 and weight 12. J. Reine Angew. Math. 494, 141–153 (1998)

    MATH  MathSciNet  Google Scholar 

  3. Braun, H.: Hermitian modular functions III. Ann. Math. 53, 143–160 (1951)

    Article  MathSciNet  Google Scholar 

  4. Cohen, D.M., Resnikoff, H.L.: Hermitian quadratic forms and Hermitian modular forms. Pac. J. Math. 76, 329–337 (1978)

    MATH  MathSciNet  Google Scholar 

  5. Conway, J., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

    MATH  Google Scholar 

  6. Conway, J., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1999)

    MATH  Google Scholar 

  7. Dern, T., Krieg, A.: Graded rings of Hermitian modular forms of degree 2. Manuscr. Math. 110, 251–272 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Feit, W.: Some lattices over \(\mathbb{Q}(\sqrt{-3})\). J. Algebra 52, 248–263 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hashimoto, K.I., Koseki, H.: Class numbers of definite unimodular Hermitian forms over the rings of imaginary quadratic number fields. Tohoku Math. J. 41, 1–30 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hentschel, M.: The Eisenstein Lattices of rank 12 over \(\mathbb{Q}(\sqrt{-3})\). http://www.mathA.rwth-aachen.de/de/mitarbeiter/hentschel/

  11. Hentschel, M.: On Hermitian theta series and modular forms. PhD thesis, RWTH Aachen (2009)

  12. Hentschel, M., Krieg, A.: A Hermitian analog of the Schottky form. In: Böcherer, S., Ibukiyama, T., Kaneko, M., Sato, F. (eds.) Automorphic Forms and Zeta Functions. Proceedings of the Conference ‘In Memory of Tsuneo Arakawa’, pp. 140–159. World Scientific, Singapore (2005)

    Google Scholar 

  13. Ikeda, T.: On the lifting of Hermitian modular forms. Compos. Math. 144, 1107–1154 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kitazume, M., Munemasa, A.: Even unimodular Gaussian lattices of rank 12. J. Number Theory 95, 77–94 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kneser, M.: Quadratische Formen. Springer, Berlin (2002)

    MATH  Google Scholar 

  16. Krieg, A.: Modular Forms on the Half-Spaces of Quaternions. Lect. Notes Math., vol. 1143. Springer, Berlin (1985)

    Google Scholar 

  17. Nebe, G., Venkov, B.: On Siegel modular forms of weight 12. J. Reine Angew. Math. 531, 49–60 (2001)

    MATH  MathSciNet  Google Scholar 

  18. Newman, M.: Integral Matrices. Academic Press, New York (1972)

    MATH  Google Scholar 

  19. Plesken, W., Souvignier, B.: Computing isometries of lattices. J. Symb. Comput. 24, 327–334 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Quebbemann, H.-G.: An application of Siegel’s formula over quaternion orders. Mathematica 31, 12–16 (1984)

    MATH  MathSciNet  Google Scholar 

  21. Schiemann, A.: Classification of Hermitian forms with the neighbour method. J. Symb. Comput. 26, 487–508 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Taylor, D.E.: The Geometry of Classical Groups. Heldermann Verlag, Berlin (1992)

    MATH  Google Scholar 

  23. Zagier, D.B.: Zetafunktionen und quadratische Zahlkörper. Springer, Berlin (1981)

    Google Scholar 

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Correspondence to Aloys Krieg.

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Communicated by U. Kühn.

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Hentschel, M., Krieg, A. & Nebe, G. On the classification of lattices over \(\mathbb{Q}(\sqrt{-3})\), which are even unimodular ℤ-lattices. Abh. Math. Semin. Univ. Hambg. 80, 183–192 (2010). https://doi.org/10.1007/s12188-010-0043-y

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