Relative Abelian Varieties If $A$ is an abelian variety, we have an addition map $\mu:A\times A\to A$.
Now, suppose we have a relative abelian variety $\mathcal{A}\to B$, i.e. the morphism is flat and proper and for any $b\in B$, $\mathcal{A}_b$ is an abelian variety.
Can we define a morphism $\mu\colon\mathcal{A}\times_B\mathcal{A}\to\mathcal{A}$ such that it restricts to the addition map on every fibre?
I think it should exist at least locally and I tried to prove its existence using base changes, but I failed.
Naively, I expect that the addition maps on the fibres glue together to form such a morphism but since I cannot find a rigorous argument I start thinking that maybe it only exists locally...?
To contextualise the question: I would like to define a relative Pontrjagin product on $\mathcal{A}$ and I was naively trying to understand if the easiest generalisation would work.
Thank you very much!
 A: I dont know if this is true in general, but I suspect it is untrue. 
However, if there is a section $e:A \to B$ such that $B_a$ is an abelian variety with identity section $e_a$ then it is true. In fact, if $A$ is connected, you only need a single fiber to be an Abelian variety. In chapter 6 of Mumford - 'Geometric Invariant Theory' one finds the following result (I am writing this from memory so there might be some details missing):
Theorem 6.14: Let $X \to S$ be a smooth proper morphism with section $e:S \to X$ where $S$ is connected and locally Noetherian. Assume that there is a geometric point $s \to S$ such that $X_s$ is an abelian variety with identity section $e_s$, then there is a (unique) multiplication $m:X \times_S X$ and $i:X \to X$ making $X/S$ into a group scheme with identity $e:S \to X$. Note that your morphism is indeed smooth since it is flat and has smooth fibers (you would need to add locally of finite presenation if your base scheme is non-Noetherian).
In general a proper smooth group scheme with connected geometric) fibers is called an Abelian scheme. Chapter 6 of Mumford's book develops the basic theory of such group schemes (for example, they are commutative and the group structure is unique once the identity section is fixed).
