The Picard group of the universal abelian variety and the Franchetta conjecture for abelian varieties

TL;DR

This paper computes the Picard group of the universal abelian variety over certain moduli stacks and proves a form of the Franchetta conjecture for these varieties over the complex numbers.

Contribution

It provides the first explicit computation of the Picard group for the universal abelian variety over moduli stacks and establishes the Franchetta conjecture in this context.

Findings

Picard group of the universal abelian variety is explicitly computed.

The Franchetta conjecture is proven for $\

The results hold over the complex numbers for moduli stacks with level structures.

Abstract

We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ 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 $\mathscr A_{g,n}$.