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Theta blocks

  • Valery Gritsenko

    Université de Lille, Villeneuve-d’Ascq, France; National Research University Higher School of Economics, Moscow, Russia

  • Nils-Peter Skoruppa

    Universität Siegen, Siegen, Germany

  • Don Zagier

    Max-Planck-Institut für Mathematik, Bonn, Germany; The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy
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Abstract

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta function and show that they give a new powerful method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

Cite this article

Valery Gritsenko, Nils-Peter Skoruppa, Don Zagier, Theta blocks. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1471