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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 4


Equivariant Gauss sum of finite quadratic forms

Shouhei Ma
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/forum-2017-0070


The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

Keywords: Gauss sum; finite quadratic forms; finite orthogonal groups; Weil representation

MSC 2010: 11L05; 11E99; 11F27


  • [1]

    B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons, New York, 1998. Google Scholar

  • [2]

    R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. CrossrefGoogle Scholar

  • [3]

    J. H. Bruinier, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780, Springer, Berlin, 2002. Google Scholar

  • [4]

    A. H. Durfee, Bilinear and quadratic forms on torsion modules, Adv. Math. 25 (1977), no. 2, 133–164. CrossrefGoogle Scholar

  • [5]

    W. Eholzer and N.-P. Skoruppa, Modular invariance and uniqueness of conformal characters, Comm. Math. Phys. 174 (1995), no. 1, 117–136. CrossrefGoogle Scholar

  • [6]

    L. J. Gerstein, Basic Quadratic Forms, Grad. Stud. Math. 90, American Mathematical Society, Providence, 2008. Google Scholar

  • [7]

    A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on 3-manifolds, Math. Ann. 253 (1980), no. 1, 29–42. CrossrefGoogle Scholar

  • [8]

    J. Milnor and P. Husemoller, Symmetric Bilinear Forms, Springer, Berlin, 1973. Google Scholar

  • [9]

    D. R. Morrison and M.-H. Saitō, Cremona transformations and degrees of period maps for K3 surfaces with ordinary double points, Algebraic Geometry (Sendai 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam (1987), 477–513. CrossrefGoogle Scholar

  • [10]

    V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238. Google Scholar

  • [11]

    N.-P. Skoruppa, Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, Bonner Math. Schriften 159, Universität Bonn, Bonn, 1985. Google Scholar

  • [12]

    C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. CrossrefGoogle Scholar

About the article

Received: 2017-04-01

Revised: 2017-11-17

Published Online: 2018-01-11

Published in Print: 2018-07-01

Citation Information: Forum Mathematicum, Volume 30, Issue 4, Pages 1029–1047, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0070.

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