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Liftings and Borcherds Products

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Book cover L-Functions and Automorphic Forms

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 10))

Abstract

This chapter serves as a brief introduction to the theory of theta-liftings with the main focus on Borcherds’ singular theta-lift and the construction of Borcherds products. Thus, after a few initial examples for liftings, we proceed to develop the tools needed to understand how the Borcherds lift works. Namely, we go through the construction of symmetric domains for orthogonal groups, introduce vector-valued modular forms and explain the definition of the Siegel theta-function. Then, we give a detailed treatment of the regularization recipe for the theta-integral and of the proof for the key properties of the additive lift: the location and type of its singularities. Finally, in the closing section, we sketch how to obtain a multiplicative lifting and the Borcherds’ products.

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Notes

  1. 1.

    If 0 happens to be a pole, yet another, slight variation of this recipe is needed, see [6].

  2. 2.

    If we want to avoid a rational weight for Ψ(z, f), we must further assume that \(c^+(0,0) \in 2{\mathbb {Z}}\). In this case, the multiplier system in i) is a character [see 6, Theorem 3.22 i)].

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Authors and Affiliations

  1. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, D-69120, Heidelberg, Germany

    Eric Hofmann

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  1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Jan Hendrik Bruinier

  2. Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

    Winfried Kohnen

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Hofmann, E. (2017). Liftings and Borcherds Products. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_19

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