Abstract
We use Kneserās 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.
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References
A. Borel, Linear Algebraic Groups, 2nd enlarged edn. Graduate Texts in Math., vol.Ā 126 (Springer, New York, 1991)
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24(3ā4), 235ā265 (1997)
S. Chisholm, Lattice methods for algebraic modular forms on quaternionic unitary groups. Ph.D. thesis, University of Calgary. Anticipated 2013
H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Math., vol.Ā 138 (Springer, Berlin, 1993)
H. Cohen, Advanced Topics in Computational Algebraic Number Theory. Graduate Texts in Math., vol.Ā 193 (Springer, Berlin, 2000)
A.M. Cohen, S.H. Murray, D.E. Taylor, Computing in groups of Lie type. Math. Comput. 73(247), 1477ā1498 (2004)
C. Cunningham, L. DembĆ©lĆ©, Computation of genus 2 Hilbert-Siegel modular forms on \(\mathbb {Q}(\sqrt{5})\) via the Jacquet-Langlands correspondence. Exp. Math. 18(3), 337ā345 (2009)
W.A. de Graaf, Constructing representations of split semisimple Lie algebras. J. Pure Appl. Algebra 164(1ā2), 87ā107 (2001). Effective methods in algebraic geometry (Bath, 2000)
L. DembĆ©lĆ©, Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comput. 76(258), 1039ā1057 (2007)
L. DembĆ©lĆ©, S. Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, in Algorithmic Number Theory. Lecture Notes in Comput. Sci., vol.Ā 5011, Banff, 2008 (Springer, Berlin, 2008), pp. 371ā386
L. DembĆ©lĆ©, J. Voight, Explicit methods for Hilbert modular forms, in Elliptic Curves, Hilbert Modular Forms and Galois Deformations (BirkhƤuser, Basel, 2013), pp.Ā 135ā198
M. Eichler, Quadratische Formen und orthogonale Gruppen (Springer, Berlin, 1952)
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.Ā 320 (Springer, Berlin, 1973), pp. 75ā151
M. Eichler, Correction to: āThe basis problem for modular forms and the traces of the Hecke operatorsā, in Modular Functions of One Variable, IV, Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972. Lecture Notes in Math., vol.Ā 476 (Springer, Berlin, 1975), pp. 145ā147
W. Fulton, J. Harris, Representation Theory: A First Course. Graduate Texts in Math., vol.Ā 129 (Springer, New York, 1991)
W.T. Gan, J.-K. Yu, Group schemes and local densities. Duke Math. J. 105(3), 497ā524 (2000)
W.T. Gan, J. Hanke, J.-K. Yu, On an exact mass formula of Shimura. Duke Math. J. 107(1), 103ā133 (2001)
B. Gross, Algebraic modular forms. Isr. J. Math. 113, 61ā93 (1999)
H. Hijikata, A.K. Pizer, T.R. Shemanske, The Basis Problem for Modular Forms on Ī 0(N) (Amer. Math. Soc., Providence, 1989)
D.W. Hoffmann, On positive definite Hermitian forms. Manuscr. Math. 71, 399ā429 (1991)
J.E. Humphreys, Linear Algebraic Groups. Graduate Texts in Math., vol.Ā 21 (Springer, New York, 1975)
K. Iyanaga, Arithmetic of special unitary groups and their symplectic representations. J. Fac. Sci. Univ. Tokyo, Sect. 1 15(1), 35ā69 (1968)
K. Iyanaga, Class numbers of definite Hermitian forms. J. Math. Soc. Jpn. 21, 359ā374 (1969)
M. Kirschmer, J. Voight, Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. 39(5), 1714ā1747 (2010)
M. Kneser, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr VerƤnderlichen. Arch. Math. 7, 323ā332 (1956)
M. Kneser, Klassenzahlen definiter quadratischer Formen. Arch. Math. 8, 241ā250 (1957)
M. Kneser, Strong approximation, in Algebraic Groups and Discontinuous Subgroups. Proc. Sympos. Pure Math., Boulder, Colo, 1965 (American Mathematical Society, Providence, 1966), pp. 187ā196
M.-A. Knus, Quadratic and Hermitian Forms over Rings (Springer, Berlin, 1991)
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.Ā 30 (Math. Soc. Japan, Tokyo, 2001), pp. 177ā195
J. Lansky, D. Pollack, Hecke algebras and automorphic forms. Compos. Math. 130(1), 21ā48 (2002)
D. Loeffler, Explicit calculations of automorphic forms for definite unitary groups. LMS J. Comput. Math. 11, 326ā342 (2008)
D. Loeffler, Computing with algebraic automorphic forms (this volume). doi:10.1007/978-3-319-03847-6_2
O.T. OāMeara, Introduction to Quadratic Forms (Springer, Berlin, 2000)
A. Pizer, An algorithm for computing modular forms on Ī 0(N). J. Algebra 64(2), 340ā390 (1980)
W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symb. Comput. 24(3ā4), 327ā334 (1997). Computational algebra and number theory (London, 1993)
W. Scharlau, Quadratic and Hermitian Forms (Springer, Berlin, 1985)
R. Scharlau, B. Hemkemeier, Classification of integral lattices with large class number. Math. Comput. 67(222), 737ā749 (1998)
A. Schiemann, Classification of Hermitian forms with the neighbour method. J. Symb. Comput. 26(4), 487ā508 (1998)
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)
G. Shimura, Arithmetic of unitary groups. Ann. Math. 79, 369ā409 (1964)
J. Socrates, D. Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves. Pac. J. Math. 219(2), 333ā364 (2005)
J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations, and L-Functions. Proc. Symp. AMS, vol.Ā 33 (1979), pp. 29ā69
M.A.A. van Leeuwen, A.M. Cohen, B. Lisser, LiE, a package for Lie group computations. CAN, Amsterdam, 1992
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.Ā 255ā298
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Greenberg, M., Voight, J. (2014). Lattice Methods for Algebraic Modular Forms on Classical Groups. In: Bƶckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_6
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DOI: https://doi.org/10.1007/978-3-319-03847-6_6
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