The Ramanujan Journal

, Volume 47, Issue 3, pp 605–650 | Cite as

Poincaré square series for the Weil representation

  • Brandon WilliamsEmail author


We calculate the Jacobi Eisenstein series of weight \(k \ge 3\) for a certain representation of the Jacobi group, and evaluate these at \(z = 0\) to give coefficient formulas for a family of modular forms \(Q_{k,m,\beta }\) of weight \(k \ge 5/2\) for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for \(Q_{k,m,\beta }\) and the Eisenstein series of Bruinier and Kuss p-adically to get an efficient algorithm. The main application is in constructing automorphic products.


Modular forms Jacobi forms Weil representation Automorphic products 

Mathematics Subject Classification

11F27 11F30 11F50 



I am grateful to Richard Borcherds, Jan Hendrik Bruinier, Sebastian Opitz, and Martin Raum for helpful discussions.


  1. 1.
    Borcherds, R.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998). ISSN 0020-9910MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borcherds, R.: The Gross–Kohnen–Zagier theorem in higher dimensions. Duke Math. J. 97(2), 219–233 (1999). ISSN 0012-7094MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruinier, J.: Borcherds products on O(2, \(l\)) and Chern classes of Heegner divisors, volume 1780 of Lecture Notes in Mathematics. Springer, Berlin (2002a). ISBN 3-540-43320-1. CrossRefGoogle Scholar
  4. 4.
    Bruinier, J.: On the rank of Picard groups of modular varieties attached to orthogonal groups. Compos. Math. 133(1), 49–63 (2002b). ISSN 0010-437XMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruinier, J., Kuss, M.: Eisenstein series attached to lattices and modular forms on orthogonal groups. Manuscr. Math. 106(4), 443–459 (2001). ISSN 0025-2611MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bykovskiĭ, V.: A trace formula for the scalar product of Hecke series and its applications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 226 (Anal. Teor. Chisel i Teor. Funktsiĭ. 13): 14–36, 235–236, 1996. ISSN 0373-2703. MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cowan, R., Katz, D., White, L.: A new generating function for calculating the Igusa local zeta function. Adv. Math. 304, 355–420 (2017). ISSN 0001-8708MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dittmann, M., Hagemeier, H., Schwagenscheidt, M.: Automorphic products of singular weight for simple lattices. Mathematische Zeitschrift 279(1), 585–603 (2015). ISSN 1432-1823MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eichler, M., Zagier, D.: The theory of Jacobi forms, volume 55 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (1985). ISBN 0-8176-3180-1. CrossRefGoogle Scholar
  10. 10.
    Igusa, J.: Complex powers and asymptotic expansions. I. Functions of certain types. J. Reine Angew. Math., 268/269:110–130 (1974). ISSN 0075-4102. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II
  11. 11.
    Raum, M.: Computing genus 1 Jacobi forms. Math. Comp. 85(298), 931–960 (2016). ISSN 0025-5718MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Scheithauer, N.: The Weil representation of \({{\rm SL}}_2(\mathbb{Z})\) and some applications. Int. Math. Res. Not. 8, 1488–1545 (2009). ISSN 1073-7928CrossRefGoogle Scholar
  13. 13.
    Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J., 58, 83–126 (1975). ISSN 0027-7630. MathSciNetCrossRefGoogle Scholar
  14. 14.
    Strömberg, F.: Weil representations associated with finite quadratic modules. Mathematische Zeitschrift 275(1), 509–527 (2013). ISSN 1432-1823MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Vol. 627, pp. 105–169. Lecture Notes in Mathematics (1977)Google Scholar
  16. 16.
    Ziegler, C.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989). ISSN 0025-5858MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

Personalised recommendations