I've been told many times that the Torelli map $J:\mathcal{M}\_g\to \mathcal{A}\_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard groups. On the level of rational picard groups, they're both generated by the determinant of the Hodge bundle, but why is this true on the nose? I'll be more than happy with a proof or with a reference to a detailed proof (actually slightly prefer the latter).



Pullback along the Torelli map is an isomorphism
I see two ways interpreting "on the nose":
1) At the level of the orbifold Picard group, or that is at stack level, and in my view this is the correct framework, both Pic\_{orb}(M\_g) and Pic\_{orb}(A\_g) are generated by the Hodge class. (The Hodge class on M\_g is by definition the pull-back of the Hodge class 
on A\_g.) The reference for this result is Arbarello and Cornalba "Picard groups of the moduli space of curves" (1987).


2) At the level of integral Picard groups, the answer is still positive but messier. 
Pic(M\_g) is the sugroup of Pic\_{orb}(M\_g) generated by the multiple m\lambda of the Hodge class which actually descends to the coarse moduli space. As far as I know, the actual value of this multiple is not known. The bundle L with class m\lambda descends to M\_g if every automorphism of a curve C, acts trivially on the fibre of L over C. It is certain to be an even number, for \lambda does not even descend to the smooth part of M\_g but 2\lambda does.


The references are Mumford's paper "Stability of proj vareties" (1977) and "Abelian quotients of Teichmüller groups" (1967). 


