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 Uspekhi Mat. Nauk, 2018, Volume 73, Issue 5(443), Pages 53–122 (Mi umn9853)

Reflective modular forms and applications

V. A. Gritsenkoabc

a Laboratoire Paul Painlevé, Université de Lille 1, Villeneuve d'Ascq, France
b Institut Universitaire de France, Paris, France
c National Research University Higher School of Economics

Abstract: The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form $\Phi_{12}$ (the generating function of the so-called Fake Monster Lie Algebra) in any of the 23 one-dimensional cusps coincides with the Kac–Weyl denominator function of the affine algebra of the root system of the corresponding Niemeier lattice. This article gives a new simple construction of the automorphic discriminant of the moduli space of Enriques surfaces as a Jacobi lifting of the product of eight theta-functions and considers three towers of reflective modular forms. One of them, the tower of $D_8$, gives a solution to a problem of Yoshikawa (2009) concerning the construction of Lorentzian Kac–Moody algebras from the automorphic discriminants connected with del Pezzo surfaces and analytic torsions of Calabi–Yau manifolds. The article also formulates some conditions on sublattices, making it possible to produce families of ‘daughter’ reflective forms from a fixed modular form. As a result, nearly 100 such functions are constructed here.
Bibliography: 77 titles.

Keywords: automorphic forms, Borcherds products, Jacobi modular forms, Kac–Moody algebras, affine Lie algebras, moduli spaces, K3-surfaces, Calabi–Yau varieties, Kodaira dimension, Hecke eigenfunctions.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 14.641.31.0001 This work was supported by the Laboratory of Mirror Symmetry, National Research University, Higher School of Economics (Russian Federation government grant, ag. no. 14.641.31.0001).

DOI: https://doi.org/10.4213/rm9853

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English version:
Russian Mathematical Surveys, 2018, 73:5, 797–864

Bibliographic databases:

UDC: 511.38+515.178+512.554.32+512.721
MSC: 11F30, 11F46, 11F50, 11F55, 14J15, 14J28, 14J33, 14J60, 14J81, 17B65, 17B67

Citation: V. A. Gritsenko, “Reflective modular forms and applications”, Uspekhi Mat. Nauk, 73:5(443) (2018), 53–122; Russian Math. Surveys, 73:5 (2018), 797–864

Citation in format AMSBIB
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• https://doi.org/10.4213/rm9853
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Gritsenko, H. Wang, “Antisymmetric paramodular forms of weight 3”, Sb. Math., 210:12 (2019), 1702–1723
2. R. Laza, K. O'Grady, “Birational geometry of the moduli space of quartic $K3$ surfaces”, Compos. Math., 155:9 (2019), 1655–1710
3. H. Wang, B. Williams, “On some free algebras of orthogonal modular forms”, Adv. Math., 373 (2020), 107332
4. V. Gritsenko, H. Wang, “Theta block conjecture for paramodular forms of weight 2”, Proc. Amer. Math. Soc., 148:5 (2020), 1863–1878
5. D. Adler, V. Gritsenko, “The d-8-tower of weak Jacobi forms and applications”, J. Geom. Phys., 150 (2020), 103616
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