\(K3\) surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space. III: The case \(r(M) \geq 18\)
type: Item
instance of: scholarly article
title: \(K3\) surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space. III: The case \(r(M) \geq 18\)
zbMATH Open document ID: 1252.14022
author: Ken-Ichi Yoshikawa
published in: Mathematische Zeitschrift
publication date: 2012-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.2841
review text: This paper establishes the automorphy of the invariant on the coarse moduli space of \(K3\) surfaces with involution, when the dimension of the moduli space is at most \(2\). The invariant was constructed earlier using equivariant analytic torsion in [Invent. Math. 156, No. 1, 53--117 (2004; Zbl 1058.58013)]. In this earlier paper, the automorphy of the invariant on the moduli space was proved when the dimension of the moduli space was \(\geq 3\). The purpose of this note is to prove the same when the dimension \(\leq 2\). Let \((X,\iota)\) be a \(K3\) surface with anti-symplectic holomorphic involution \(\iota\). Let \(M\) be a sublattice of the \(K3\) lattice \(H_2(X,{\mathbb{Z}})\), and let \(M^{\perp}\) be its orthogonal complement. \((X, \iota)\) is called a \(2\)-elementary of type \(M\) if \(H_+^2(X,{\mathbb{Z}})\) (=the invariant sublattice of \(H^2(X,{\mathbb{Z}})\) with respect to the action of \(\iota\)) is isometric to \(M\). The period domain \(\Omega_{M^{\perp}}^+\) of \(2\)-elementary \(K3\) surfaces of type \(M\) is isomorphic to a symmetric domain of type IV of dimension \(20-r(M)\) where \(r(M)\) denotes the rank of \(M\). When \(r(M)\leq 17\), the automorphic property of the function \(\tau_M\) was proved in the earlier paper of the author mentioned above. In this paper, the author considers the case \(r(M)\geq 18\), and extends the automorphic property of the invariant \(\tau_M(X,\iota)\) in this case. The main result is formulated in the following Theorem. There exists an integer \(\nu\in {\mathbb{Z}}_{>0}\), an (possibly meromorphic) automorphic form \(\Psi_M\) on \(\Omega_{M^{\perp}}^+\) of weight \(\nu(r(M)-6)\) and a Siegel modular form \(S_M\) on the Siegel upper-half space \({\mathcal{S}}_{g(M)}\) of weight \(4\nu\) such that, for every \(2\)-elementary \(K3\) surface \((X,\iota)\) of type \(M\), the invariant \[ \tau_M(X,\iota)=\|\Psi_M(\overline{\omega}_M(X,\iota)\|^{-1/2\nu} \|S_M(\Omega(X^{\iota}\|^{-1/2\nu}. \] Here \(\overline{\omega…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 6097142
zbMATH Keywords: \(K3\) surface with involution
zbMATH Keywords: analytic torsion
zbMATH Keywords: automorphic form
MaRDI profile type: Publication
OpenAlex ID: W2021854726
arXiv ID: 1007.2841
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DOI: 10.1007/S00209-011-0928-1
description: scientific article; zbMATH DE number 6097142