My question concerns the Picard-Functor for abelian schemes, as it is for example introduced in the book of Birkenhake and Lange "Complex Abelian Varieties" in Appendix F.

So let $X\rightarrow S$ be an abelian scheme over a locally noetherian basis $S$. Let $\mathcal L$ be a line bundle on $X$. It is called rigidified along the zero section $e$ of $X$ if one has an isomorphism $e^*L \simeq \mathcal O_S$.

For each locally noetherian $T\rightarrow S$ they now define the following abelian group

$Pic_{X/S}(T)$:= Isomorphism classes of line bundles on $X\times_S T$ rigidified along the zero section of $X\times_S T$.

Then they say that the resulting functor is representable by the Picard scheme etc. Similar discussion for $Pic^0$.

Now my question:

What do they exactly mean? Do they mean:

(1) the group of iso classes of line bundles, which additionally have the property of being rigidified; an iso here just an iso of line bundles (though both have the addition of being rigidified)

or

(2) the group of iso classes of pairs $(\mathcal L, \phi)$, where $\mathcal L$ is a line bundle on $X\times_S T$, and $\phi$ is a rigidification of it. Here an isomorphism of such a pair is defined in the obvious way.

Addendum: Even with the answers below one point still isn't clear to me:

Take an abelian variety $X$ over a field $k$. Then Mumford defines the dual $\hat{X}$ of it by saying that $Hom_k(S,\hat{X}) \simeq$ Isoclasses of line bundles on $X\times S$ which (i) are in $Pic^0(X)$ fibrewise with respect to fibers of $S$ and (!) (ii) $\mathcal L$ is trivial on $0\times S$. This definitely doesn't look like (2), but rather (1). My problem is that if he meant (2), then for the same bundle satisfying (i) and (ii) one would get different morphisms $S \rightarrow \hat{X}$, as one can change the rigidification always by a nonzero scalar in $k$. But he always uses the dual of $X$ as if just the data (i) and (ii) would induce a unique morphism $S\rightarrow \hat{X}$.

To sum it up: my problem is that in the case (2) one gets much more morphisms to the representing object than in the other case (1), and this can't be true I think. So where is my fault?