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The Ramanujan Journal

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

Poincaré square series for the Weil representation

  • Brandon Williams
Article

Abstract

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.

Keywords

Modular forms Jacobi forms Weil representation Automorphic products 

Mathematics Subject Classification

11F27 11F30 11F50 

Notes

Acknowledgements

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

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Copyright information

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

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA

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