Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 108, 17 pages      arXiv:1810.02048
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Hyper-Algebras of Vector-Valued Modular Forms

Martin Raum
Chalmers tekniska högskola och Göteborgs Universitet, Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden

Received May 07, 2018, in final form September 30, 2018; Published online October 04, 2018

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.

Key words: Siegel modular forms; vector-valued Hecke operators; automorphic representations.

pdf (455 kb)   tex (27 kb)


  1. Furusawa M., Morimoto K., On special Bessel periods and the Gross-Prasad conjecture for ${\rm SO}(2n+1)\times{\rm SO}(2)$, Math. Ann. 368 (2017), 561-586, arXiv:1611.05567.
  2. Hart W., Johansson F., Pancratz S., FLINT: Fast Library for Number Theory, v2.5.2 ed., 2015, available at
  3. Helgason S., Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. 12XII, Academic Press, New York - London, 1962.
  4. Holt D.F., Eick B., O'Brien E.A., Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005.
  5. Imamoglu O., Kohnen W., Representations of integers as sums of an even number of squares, Math. Ann. 333 (2005), 815-829.
  6. Klemm A., Poretschkin M., Schimannek T., Westerholt-Raum M., Direct integration for genus two mirror curves, arXiv:1502.00557.
  7. Kohnen W., Skoruppa N.-P., A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95 (1989), 541-558.
  8. Kohnen W., Zagier D., Modular forms with rational periods, in Modular Forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, 197-249.
  9. Krieg A., Hecke algebras, Mem. Amer. Math. Soc. 87 (1990), x+158 pages.
  10. Krieg A., Raum M., The functional equation for the twisted spinor $L$-series of genus 2, Abh. Math. Semin. Univ. Hambg. 83 (2013), 29-52, arXiv:0907.2767.
  11. Marks C., Mason G., Structure of the module of vector-valued modular forms, J. Lond. Math. Soc. 82 (2010), 32-48, arXiv:0901.4367.
  12. PARI/GP, Version 2.7.3, 2015, available at
  13. Pitale A., Saha A., Schmidt R., Representations of $\mathrm{SL}_2(\mathbb{R})$ and nearly holomorphic modular forms, arXiv:1501.00525.
  14. Rankin R.A., The scalar product of modular forms, Proc. London Math. Soc. 2 (1952), 198-217.
  15. Raum M., Efficiently generated spaces of classical Siegel modular forms and the Böcherer conjecture, J. Aust. Math. Soc. 89 (2010), 393-405, arXiv:1002.3883.
  16. Sage Mathematics Software, Version 6.9, 2015, available at
  17. Shimura G., Nearly holomorphic functions on Hermitian symmetric spaces, Math. Ann. 278 (1987), 1-28.
  18. Sullivan J.B., Representations of the hyperalgebra of an algebraic group, Amer. J. Math. 100 (1978), 643-652.
  19. Taylor K., Analytic continuation of nonanalytic vector-valued Eisenstein series, Ph.D. Thesis, Temple University, 2006.
  20. Waldspurger J.-L., Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484.
  21. Wall H.S., Hypergroups, Amer. J. Math. 59 (1937), 77-98.
  22. Wallach N.R., Real reductive groups. I, Pure and Applied Mathematics, Vol. 132, Academic Press, Inc., Boston, MA, 1988.
  23. Westerholt-Raum M., Products of vector valued Eisenstein series, Forum Math. 29 (2017), 157-186, arXiv:1411.3877.
  24. Westerholt-Raum M., Harmonic weak Siegel-Maaß forms I: preimages of non-holomorphic Saito-Kurokawa lifts, Int. Math. Res. Not. 2018 (2018), 1442-1472, arXiv:1510.03342.

Previous article  Next article   Contents of Volume 14 (2018)