

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conjecture.


Introduction
In [39], a holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. The purpose of the present paper is to completely determine the structure of this invariant as a function on the moduli space of such K3 surfaces. We shall express it using Borcherds products and Siegel modular forms. Let us explain our result in detail.
A pair consisting of a K3 surface and its anti-symplectic involution is called a 2-elementary K3 surface. By Nikulin [29], the deformation type of a 2-elementary In [39], a holomorphic torsion invariant of 2-elementary K3 surfaces was defined as follows. Let (X, ι) be a 2-elementary K3 surface of type M. Write X ι for the set of fixed points of ι. Depending on M, X ι is either empty or the disjoint union of smooth curves of total genus g = g(M ). (See §3.2 for a formula for g.) When X ι is empty, the corresponding type is unique and is called exceptional, in which case (X, ι) is the universal covering of an Enriques surface endowed with the non-trivial covering transformation. Take an ι-invariant Kähler form γ on X and a holomorphic 2-form η = 0 on X. Let τ Z2 (X, γ)(ι) be the equivariant analytic torsion of (X, γ) with respect to the ι-action and let τ (X ι, γ| X ι ) be the analytic torsion of (X ι, γ| X ι ) (cf. [30], [6]). Then the real number τ M (X, ι) = Vol(X, γ) 14−r 4 τ Z2 (X, γ)(ι)Vol(X ι, γ| X ι )τ (X ι, γ| X ι ) depends only on the isomorphism class of (X, ι), so that it gives rise to a function τ M on M 0 Λ. The goal of this paper is to give an explicit expression of τ M in terms of modular forms. It turns out that τ M is expressed by two types of modular forms: Borcherds products and Siegel modular forms. Let us explain these modular forms.
Let ρ Λ : Mp 2 (Z) → GL(C[A Λ ]) be the Weil representation attached to the discriminant group A Λ of Λ, where C[A Λ ] is the group ring of A Λ. By Borcherds [8], given an O + (Λ)-invariant elliptic modular form f of type ρ Λ and of weight 1−(20−r)/2 with integral Fourier expansion, we can take its Borcherds lift Ψ Λ (·, f ). This is a (possibly meromorphic) automorphic form for O + (Λ) with infinite product expansion, and its Petersson norm Ψ Λ (·, f ) descends to a function on M 0 Λ. To express τ M, the Borcherds lift of the following elliptic modular form shall be used. Let η(τ ) be the Dedekind η-function and let θ A + 1 (τ ) be the theta series of the A 1 -lattice. We put This induces the following modular form of type ρ Λ (cf. [9], [35], [40]): where | γ is the Petersson slash operator and e 0 ∈ C[A Λ ] is the vector corresponding to 0 ∈ A Λ. Except for two types, the Borcherds lift of (2 g−1 + δ r,10 )F Λ shall be used in the expression of τ M, where δ i,j denotes the Kronecker delta.
On the other hand, Siegel modular forms also yield functions on M 0 Λ. The period map for the fixed curves of 2-elementary K3 surfaces induces a holomorphic map J M : M 0 Λ → A g, where A g is the Siegel modular variety of degree g. Then the pullback of the Petersson norm of a Siegel modular form by J M is a function on M 0 Λ. The following where Σ runs over all ineffective theta characteristics on X ι and τ (X ι, Σ; γ| X ι ) is the analytic torsion of Σ with respect to γ| X ι. It turns out that τ spin M (X, ι) is independent of the choice of an ι-invariant Kähler form γ and gives rise to a function τ spin M on M 0 Λ. Our second result is stated as follows (cf. Theorem 10.2): Theorem 0.2. Let H Λ ⊂ M Λ be the characteristic Heegner divisor (cf. Section 2). Then there exists a constant C ′ M > 0 depending only on M such that the following equality of functions on M 0 Λ \ H Λ holds: We remark that if H Λ = ∅, then τ spin M jumps along H Λ and thus τ spin M is a discontinuous function on M 0 Λ. For an explicit relation between the constants in Theorems 0.1 and 0.2, see Section 10.2 below. After Theorem 0.2, it is very likely that C ′ M can be determined up to an algebraic number by an Arakelov geometric study of 2-elementary K3 surfaces with maximal Picard number. This subject shall be discussed elsewhere. According to Theorem 0.2, the invariant τ spin M is elliptic modular in the sense that it is the Borcherds lift of an elliptic modular form. A conceptual account for this unexpected elliptic modularity as well as the geometric meaning of the corresponding modular forms is strongly desired.
The divisor of Ψ Λ (·, 2 g−1 F Λ ) is calculated by the theory of Borcherds product [8], so the problem is reduced to calculating the divisor of J * M χ 8 g. Since δ = 1, the isometry class of M is determined by g and the number k + 1 of the components of the fixed curve. (See Section 3.2 for the formula for the invariants g = g(M ) and k = k(M ).) Write M g,k for the lattice with these invariants and set Λ g,k = M ⊥ g,k. Let D Λ g,k = D + Λ g,k + D − Λ g,k be the decomposition according to the type of (−2)-vectors of Λ g,k, and let H Λ g,k be the characteristic Heegner divisor (cf. Section 2). We will proceed inductively on k.
(b) We first study the series k = 0 by a geometric approach (Section 6). Curves with vanishing theta constants are characterized by the existence of certain halfcanonical bundle. By analyzing bi-anticanonical sections of Del Pezzo surfaces with such property, we prove that the support of div(J * Mg,0 χ 8 g ) on M 0 Λg,0 coincides with H Λg,0. Hence there exist integers a g, b g, c g ≥ 0 such that (0. 4) div(J * Mg,0 χ 8 g ) = a g D − Λg,0 + b g H Λg,0 + c g D + Λg,0. (c) We have a natural inclusion of lattices Λ g,k+1 ⊂ Λ g,k, which induces an inclusion of domains i : Ω Λ g,k+1 ֒→ Ω Λ g,k. Then we show that J M g,k • i = J M g,k+1 outside a locus of codimension 2 and that i * H Λ g,k = 2H Λ g,k+1 and similar relation between D ± Λ g,k and D ± Λ g,k+1 (cf. Sections 2 and 3). This enables us to inductively extend (0.4) to the case k ≥ 1 (Section 8): div(J * M g,k χ 8 g ) = a g,k D − Λ g,k + b g,k H Λ g,k + c g,k D + Λ g,k, where a g,k, b g,k, c g,k are integers satisfying a g,k = a g, b g,k = 2 k b g, c g,k = 0 (g < 10).
By using a formula in [40] and the formula [36] for the thetanull divisor on the moduli space of curves, we also prove the estimates a g ≥ 2 2g−1 and b g ≥ 2 4. (d) Substituting these relations and estimates in the left-hand side of (0.3), we obtain the desired inequality when g < 10. In case g = 10, an extra argument is required. See Section 8. Note