Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors
type: Item
instance of: scholarly article
title: Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors
zbMATH Open document ID: 1004.11021
published in: Lecture Notes in Mathematics
publication date: 2002-06-04T00:00:00Z
full work available at URL: t1780.htm
review text: Let \((V,q)\) be a quadratic space of signature \((2,\ell)\), and let \(L \subset V\) be an even unimodular lattics, which implies that \(\ell\) is even. In a pioneering paper, \textit{R. Borcherds} [Invent. Math. 120, 161-213 (1995; Zbl 0932.11028)] discovered a lifting from nearly holomorphic modular forms of weight \(1-\ell/2\) for \(SL(2,\mathbb Z)\) to meromorphic modular forms for the orthogonal group of \(L\), which have infinite product expansions analogous to the Dedekind eta-function. This book discusses geometric aspects of the Borcherds lifting. Let \(O^+ (V)\) be the connected component of the identity of the orthogonal group \(O(V)\) of \(V\), and let \(H\) be a maximal compact subgroup of \(O^+ (V)\). Then the quotient space \(O^+ (V) /H\) can be realized as a tube domain \(\mathbb H_\ell\) in \(\mathbb C^\ell\). The orthogonal group \(O(L)\) of the lattice \(L\) is an arithmetic subgroup of \(O(V)\), and the subgroup \(\Gamma (L) = O(L) \cap O^+ (V)\) acts on \(\mathbb H_\ell\). The associated quotient space \(\mathcal X_L = \mathbb H_L \backslash \Gamma (L)\) has the structure of a quasi-projective variety over \(\mathbb C\). To each nonnegative integer \(m\), one can associate a \(\Gamma (L)\)-invariant divisor \(H(m)\), known as the Heegner divisor of discriminant \(m\). The Borcherds lifting determines explicit relations between Heegner divisors in the divisor class group \(\text{Cl} (\mathcal X_L)\) of \(\mathcal X_L\), which can be used for the study of the geometry of \(\mathcal X_L\). In fact, it is more convenient to work with a slightly modified divisor group \(\widetilde{\text{Cl}} (\mathcal X_L)\), which is defined as the quotient of \(\text{Cl} (\mathcal X_L)\) by the subgroup of divisors coming from meromorphic modular forms for \(\Gamma (L)\) of rational weight with some multiplier system. Let \(\kappa = 1+\ell/2\), and denote by \(S_\kappa\) the space of cusp forms of weight \(\kappa\) for \(SL(2,\mathbb Z)\). Let \(\mathcal A_\kappa …
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F23
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 1749034
zbMATH Keywords: automorphic forms
zbMATH Keywords: theta series
zbMATH Keywords: Borcherds products
zbMATH Keywords: Heegner divisors
zbMATH Keywords: Chern classes
zbMATH Keywords: Borcherds lifting
author: Jan Hendrik Bruinier
MaRDI profile type: Publication
OpenAlex ID: W313412277
DOI: 10.1007/B83278
description: scientific article; zbMATH DE number 1749034