The Picard Group of the Universal Abelian Variety and the Franchetta Conjecture for Abelian Varieties

Abstract

We compute the Picard group of the universal Abelian variety over the moduli stack ${\mathcal{A}}_{g,n}$ of principally polarized Abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for ${\mathcal{A}}_{g,n}$.