Abstract
Gritsenko, Skoruppa and Zagier associated to a root system R a theta block \(\vartheta _R\), which is a Jacobi form of lattice index. We classify the theta blocks \(\vartheta _R\) of q-order 1 and show that their Gritsenko lift is a strongly-reflective Borcherds product of singular weight, which is related to Conway’s group \({\text {Co}}_0\). As a corollary we obtain a proof of the theta block conjecture by Gritsenko, Poor and Yuen for the pure theta blocks obtained as specializations of the functions \(\vartheta _R\).
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Acknowledgements
We thank Nils Scheithauer for helpful discussions on the content of this paper and for providing us his unpublished notes [15]. H. Wang would like to thank Valery Gritsenko, Nils-Peter Skoruppa and Brandon Williams for many helpful discussions, and he is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support where this work was done. H. Wang was supported by the Institute for Basic Science (IBS-R003-D1).
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Dittmann, M., Wang, H. Theta blocks related to root systems. Math. Ann. 384, 1157–1180 (2022). https://doi.org/10.1007/s00208-021-02316-1
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DOI: https://doi.org/10.1007/s00208-021-02316-1