Determinant Bundles for Abelian Schemes

To a symmetric, relatively ample line bundle on an abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that p ≡ −1mod(4) and p ≤ 2g − 1, where g is the relative dimension of the abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack. Let L be a relatively ample line bundle on an abelian scheme π : A→ S, trivialized along the zero section. Assume that L is symmetric, i. e. [−1]∗AL ≃ L. We denote by φL : A→  the corresponding self-dual homomorphism (where  is the dual abelian scheme). Let d = rkπ∗L, so that d 2 is the degree of φL. Then Faltings and Chai proved in [4] the following equality in Pic(S): 8 · d · det(π∗L) = −4 · d 4 · ωA where we denote by ωA the restriction of the relative canonical bundle ωA/S to the zero section. In other words, the element ∆(L) := 2 · det(π∗L) + d · ωA of Pic(S) is annihilated by 4d. It is known from the transformation theory of theta-functions (see [10]) that this result is sharp for principal polarizations (d = 1). In the case of analitic families of complex abelian varieties A. Kouvidakis showed in [7] using theta functions that if the type of polarization is (d1, . . . , dg) with d1| . . . |dg then 4 ·∆(L) = 0 except when 3|dg and dg−1 6≡ 0mod(3). In the latter case he proved that 12 · ∆(L) = 0. This suggests that one can try to eliminate the factor d in general situation. We prove that one can do this outside certain set of prime divisors of d. In particular we explain the appearence of factor 3 above algebraically (see Theorem 1.1). Here are the precise statements. Theorem 0.1. Let L be a symmetric, relatively ample line bundle over an abelian scheme A/S of relative dimension g, trivialized along the zero section. Then (1) 22 · d ·∆(L) = 0 where d = ∏ pp is the product of powers of primes p dividing d such that