A conjectured analogue of Dedekind's eta function for \(K3\) surfaces (Q1902229)

For \(\tau\in \mathbb{C}\) let \(E_\tau =\mathbb{C}/ \Lambda_\tau\) be the associated elliptic curve; the Dedekind's eta function \(\eta (\tau)\) provides a formula for the product of the non 0 eigenvalues of the Laplacian \(\Delta= -4b \partial^2/ \partial z \partial \overline z\). This theory has been extended to K3 surfaces, by means of a function \(f_e\) defined in the space of triples \((X, \alpha,e)\) with \(X,e\) polarized K3 surface and \(\alpha\) basis for \(H_2(X,Z)\); \(f_e\) is connected with the product of non 0 eigenvalues of the Laplacian \(\Delta^{(0,1)}\) acting on smooth (0,1)-forms. In the present paper, the authors go through the construction of \(f_e\) and provide some evidence that it has a product formula and other properties analogue to the properties of \(\eta (\tau)\).