\(K3\) surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space. III: The case \(r(M) \geq 18\)
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 \(\overline{\omega}_M(X,\iota)\) denotes the period of \((X,\iota)\), \(\Omega(X^{\iota})\) the period of \(X^{\iota}\), and \(\|\cdot\|\) denotes the Petersson norm.
scientific article; zbMATH DE number 6097142
scientific article