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 ${\mathcal{A}}_{g,n}$.

17 pages