Lattice Methods for Algebraic Modular Forms on Classical Groups

Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 6)

Abstract

We use Kneser’s neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups.

References

  1. [Bor91]
    A. Borel, Linear Algebraic Groups, 2nd enlarged edn. Graduate Texts in Math., vol. 126 (Springer, New York, 1991) CrossRefMATHGoogle Scholar
  2. [BCP97]
    W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997) CrossRefMATHMathSciNetGoogle Scholar
  3. [Chi]
    S. Chisholm, Lattice methods for algebraic modular forms on quaternionic unitary groups. Ph.D. thesis, University of Calgary. Anticipated 2013 Google Scholar
  4. [Coh93]
    H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Math., vol. 138 (Springer, Berlin, 1993) CrossRefMATHGoogle Scholar
  5. [Coh00]
    H. Cohen, Advanced Topics in Computational Algebraic Number Theory. Graduate Texts in Math., vol. 193 (Springer, Berlin, 2000) CrossRefGoogle Scholar
  6. [CMT04]
    A.M. Cohen, S.H. Murray, D.E. Taylor, Computing in groups of Lie type. Math. Comput. 73(247), 1477–1498 (2004) CrossRefMATHMathSciNetGoogle Scholar
  7. [CD09]
    C. Cunningham, L. Dembélé, Computation of genus 2 Hilbert-Siegel modular forms on \(\mathbb {Q}(\sqrt{5})\) via the Jacquet-Langlands correspondence. Exp. Math. 18(3), 337–345 (2009) CrossRefMATHGoogle Scholar
  8. [dGr01]
    W.A. de Graaf, Constructing representations of split semisimple Lie algebras. J. Pure Appl. Algebra 164(1–2), 87–107 (2001). Effective methods in algebraic geometry (Bath, 2000) CrossRefMATHMathSciNetGoogle Scholar
  9. [Dem07]
    L. Dembélé, Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comput. 76(258), 1039–1057 (2007) CrossRefMATHGoogle Scholar
  10. [DD08]
    L. Dembélé, S. Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, in Algorithmic Number Theory. Lecture Notes in Comput. Sci., vol. 5011, Banff, 2008 (Springer, Berlin, 2008), pp. 371–386 CrossRefGoogle Scholar
  11. [DV]
    L. Dembélé, J. Voight, Explicit methods for Hilbert modular forms, in Elliptic Curves, Hilbert Modular Forms and Galois Deformations (Birkhäuser, Basel, 2013), pp. 135–198 CrossRefGoogle Scholar
  12. [Eic52]
    M. Eichler, Quadratische Formen und orthogonale Gruppen (Springer, Berlin, 1952) CrossRefMATHGoogle Scholar
  13. [Eic73]
    M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, in Modular Functions of One Variable, I. Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972. Lecture Notes in Math., vol. 320 (Springer, Berlin, 1973), pp. 75–151 CrossRefGoogle Scholar
  14. [Eic75]
    M. Eichler, Correction to: “The basis problem for modular forms and the traces of the Hecke operators”, in Modular Functions of One Variable, IV, Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972. Lecture Notes in Math., vol. 476 (Springer, Berlin, 1975), pp. 145–147 Google Scholar
  15. [FH91]
    W. Fulton, J. Harris, Representation Theory: A First Course. Graduate Texts in Math., vol. 129 (Springer, New York, 1991) CrossRefMATHGoogle Scholar
  16. [GY00]
    W.T. Gan, J.-K. Yu, Group schemes and local densities. Duke Math. J. 105(3), 497–524 (2000) CrossRefMATHMathSciNetGoogle Scholar
  17. [GHY01]
    W.T. Gan, J. Hanke, J.-K. Yu, On an exact mass formula of Shimura. Duke Math. J. 107(1), 103–133 (2001) CrossRefMATHMathSciNetGoogle Scholar
  18. [Gro99]
    B. Gross, Algebraic modular forms. Isr. J. Math. 113, 61–93 (1999) CrossRefMATHGoogle Scholar
  19. [HPS89]
    H. Hijikata, A.K. Pizer, T.R. Shemanske, The Basis Problem for Modular Forms on Γ 0(N) (Amer. Math. Soc., Providence, 1989) Google Scholar
  20. [Hof91]
    D.W. Hoffmann, On positive definite Hermitian forms. Manuscr. Math. 71, 399–429 (1991) CrossRefMATHGoogle Scholar
  21. [Hum75]
    J.E. Humphreys, Linear Algebraic Groups. Graduate Texts in Math., vol. 21 (Springer, New York, 1975) CrossRefMATHGoogle Scholar
  22. [Iya68]
    K. Iyanaga, Arithmetic of special unitary groups and their symplectic representations. J. Fac. Sci. Univ. Tokyo, Sect. 1 15(1), 35–69 (1968) MATHMathSciNetGoogle Scholar
  23. [Iya69]
    K. Iyanaga, Class numbers of definite Hermitian forms. J. Math. Soc. Jpn. 21, 359–374 (1969) CrossRefMATHMathSciNetGoogle Scholar
  24. [KV]
    M. Kirschmer, J. Voight, Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. 39(5), 1714–1747 (2010) CrossRefMATHMathSciNetGoogle Scholar
  25. [Kne56]
    M. Kneser, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen. Arch. Math. 7, 323–332 (1956) CrossRefMATHMathSciNetGoogle Scholar
  26. [Kne57]
    M. Kneser, Klassenzahlen definiter quadratischer Formen. Arch. Math. 8, 241–250 (1957) CrossRefMATHMathSciNetGoogle Scholar
  27. [Kne66]
    M. Kneser, Strong approximation, in Algebraic Groups and Discontinuous Subgroups. Proc. Sympos. Pure Math., Boulder, Colo, 1965 (American Mathematical Society, Providence, 1966), pp. 187–196 CrossRefGoogle Scholar
  28. [Knu91]
    M.-A. Knus, Quadratic and Hermitian Forms over Rings (Springer, Berlin, 1991) CrossRefMATHGoogle Scholar
  29. [Koh01]
    D. Kohel, Hecke module structure of quaternions, in Class Field Theory: Its Centenary and Prospect, Tokyo, 1998, ed. by K. Miyake. Adv. Stud. Pure Math. vol. 30 (Math. Soc. Japan, Tokyo, 2001), pp. 177–195 Google Scholar
  30. [LP02]
    J. Lansky, D. Pollack, Hecke algebras and automorphic forms. Compos. Math. 130(1), 21–48 (2002) CrossRefMATHMathSciNetGoogle Scholar
  31. [Loe08]
    D. Loeffler, Explicit calculations of automorphic forms for definite unitary groups. LMS J. Comput. Math. 11, 326–342 (2008) CrossRefMATHMathSciNetGoogle Scholar
  32. [Loe]
    D. Loeffler, Computing with algebraic automorphic forms (this volume). doi:10.1007/978-3-319-03847-6_2
  33. [O’Me00]
    O.T. O’Meara, Introduction to Quadratic Forms (Springer, Berlin, 2000) MATHGoogle Scholar
  34. [Piz80]
    A. Pizer, An algorithm for computing modular forms on Γ 0(N). J. Algebra 64(2), 340–390 (1980) CrossRefMATHMathSciNetGoogle Scholar
  35. [PS97]
    W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symb. Comput. 24(3–4), 327–334 (1997). Computational algebra and number theory (London, 1993) CrossRefMATHMathSciNetGoogle Scholar
  36. [Sch85]
    W. Scharlau, Quadratic and Hermitian Forms (Springer, Berlin, 1985) CrossRefMATHGoogle Scholar
  37. [SH98]
    R. Scharlau, B. Hemkemeier, Classification of integral lattices with large class number. Math. Comput. 67(222), 737–749 (1998) CrossRefMATHMathSciNetGoogle Scholar
  38. [Schi98]
    A. Schiemann, Classification of Hermitian forms with the neighbour method. J. Symb. Comput. 26(4), 487–508 (1998) CrossRefMATHMathSciNetGoogle Scholar
  39. [Schu91]
    R. Schulze-Pillot, An algorithm for computing genera of ternary and quaternary quadratic forms, in Proc. Int. Symp. on Symbolic and Algebraic Computation, Bonn (1991) Google Scholar
  40. [Shi64]
    G. Shimura, Arithmetic of unitary groups. Ann. Math. 79, 369–409 (1964) CrossRefMATHMathSciNetGoogle Scholar
  41. [SW05]
    J. Socrates, D. Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves. Pac. J. Math. 219(2), 333–364 (2005) CrossRefMATHMathSciNetGoogle Scholar
  42. [Tit79]
    J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations, and L-Functions. Proc. Symp. AMS, vol. 33 (1979), pp. 29–69 CrossRefGoogle Scholar
  43. [vLCL92]
    M.A.A. van Leeuwen, A.M. Cohen, B. Lisser, LiE, a package for Lie group computations. CAN, Amsterdam, 1992 Google Scholar
  44. [Voi]
    J. Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, in Quadratic and Higher Degree Forms. Developments in Math., vol. 31 (Springer, New York, 2013), pp. 255–298 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of CalgaryCalgaryCanada
  2. 2.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA
  3. 3.Department of MathematicsDartmouth CollegeHanoverUSA

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