K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M)>10

TL;DR

This paper investigates the structure of invariants of K3 surfaces with involution, revealing that for certain lattice ranks, the associated automorphic form decomposes into a product of a Borcherds lift and a Siegel modular form, advancing understanding of their moduli space.

Contribution

It provides a structure theorem showing the automorphic form associated with K3 surfaces with involution decomposes into known modular forms when the lattice rank exceeds 10.

Findings

Automorphic form expressed as tensor product of Borcherds lift and Igusa's Siegel modular form for r(M)>10.

Proves a new structure theorem for invariants of K3 surfaces with involution.

Enhances understanding of the moduli space of K3 surfaces with involution.

Abstract

We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to the involution is strictly bigger than 10, we prove that this automorphic form is expressed as the tensor product of an explicit Borcherds lift and Igusa's Siegel modular form.