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

This article is cited in 5 scientific papers (total in 5 papers)

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).


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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
Received: 14.08.2018

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
\by V.~A.~Gritsenko
\paper Reflective modular forms and applications
\jour Uspekhi Mat. Nauk
\yr 2018
\vol 73
\issue 5(443)
\pages 53--122
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 5
\pages 797--864

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    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. R. Laza, K. O'Grady, “Birational geometry of the moduli space of quartic $K3$ surfaces”, Compos. Math., 155:9 (2019), 1655–1710  crossref  mathscinet  zmath  isi
    3. H. Wang, B. Williams, “On some free algebras of orthogonal modular forms”, Adv. Math., 373 (2020), 107332  crossref  mathscinet  zmath  isi
    4. V. Gritsenko, H. Wang, “Theta block conjecture for paramodular forms of weight 2”, Proc. Amer. Math. Soc., 148:5 (2020), 1863–1878  crossref  mathscinet  zmath  isi
    5. D. Adler, V. Gritsenko, “The d-8-tower of weak Jacobi forms and applications”, J. Geom. Phys., 150 (2020), 103616  crossref  mathscinet  zmath  isi
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