Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other cases it has order at most 4 [Silverman, AEC VII.6.1]. I'm interested to know what you can say about how the size of the component group varies in a family of elliptic curves (and more generally in a family of abelian varieties). Let's look at an example: consider the family $E\subset \mathbb{P}^2 \times \operatorname{Spec}(\mathbb{Z}[a,b][1/(4a^3+27b^2)])$ cut out by $Y^2Z = X^3+aX^2Z+bZ^3$ (where $S=\operatorname{Spec}(\mathbb{Z}[a,b][1/(4a^3+27b^2)])$). On the compact subset of $S(\mathbb{Q}_p)$ where $v(2^8*(3a)^3/(4a^3+27b^2))\geq i$ (for any integer $i$), the result quoted above tells us that the sizes of the component groups of the Néron models of the elliptic curves that lie over this compact subset are uniformly bounded by $\max\{-i, 4\}$ (since we only have multiplicative reduction when the j-invariant has negative valuation). The key point here is that the size of the component group depends on the valuation of some rational function of the parameters (in this case, the j-invariant). So in some sense I think of the size of the component groups as "varying continuously" in the family.
This leads me to the following question: Given a family of abelian varieties $\mathcal{A}\rightarrow S$ with $S$ quasi-projective over $\operatorname{Spec}(\mathbb{Z})$ and a compact subset $C\subset S(\mathbb{Q}_p)$ (in the induced $p$-adic topology), is there a uniform bound on the size of the components groups of the Néron models of the fibers $\mathcal{E}_c$ for $c\in C$? I believe this should follow from some variation of a "rigid uniformization", but I'm not fluent enough in the world of rigid geometry to know if this result is well-known.
